Looking to get yourself motivated? We've got something to help you kickstart yourself into action in the form of some recommended reading for all you aspiring animators!
One of the many highlights of this book are the interviews conducted with a slew of professional animators, Ken Duncan, (Beauty and the Beast, Aladdin, 9), Jason Figliozzi, (Wreck it Ralph, Frozen, Big Hero 6) and T. Dan Hofstedt, (Pocahontas, Mulan, Planes), to name but a few, which help to reinforce the principles and processes explained throughout the book!
Created by two of the famed Nine Old Men (The Walt Disney Company's core animators), this book is a must for anyone who appreciates animation, and Disney animation in particular. Detailing Disney's process for completing a picture step by step, the book provides indepth information for animators covering the history of how animation came to be as well as important animation concepts.
California native Preston Blair (Fantasia) is somewhat of an institution amongst our 2D artists. His animation collection series provide a thorough basis for studying animation principles and techniques, you'll learn how to develop a character, create dynamic movement, and animate dialogue with action. This is a book that can easily be reffered back to throughout your career.
You'll find this book on many an animator's book shelf! Extensively covering timing sheets, lip-sync timing and cycles - this book offers great help for those who may struggle with space and timing in their animation. The second edition includes timing for digital production, digital storyboarding in 2D and 3D, the use of After Effects and more!
Ed Hook is an industry leading acting instructor for animators providing Masterclasses worldwide. He uses basic acting theory to help guide animators in creating their characters in a realistic way. The book provides a plethora of acting theory, examples and exercises to help you develop the acting skills needed to breathe life into your animated characters.
So there you have it - our staff's #TopPicks for your animation reading list!
Have you got any animation book recommendations we're missing? Pop them in the comments below and help us grow our list!


New Port Richey, Florida – Last November, two men accused of running two prescription drug rings were arrested after a two-year investigation allegedly revealed the men and their suspected crews had shipped tens of thousands of oxycodone pills from Florida to Tennessee and Kentucky.
On Friday, U.S. Attorney Robert E. O'Neill announced the unsealing of two federal indictments charging 31 suspected crew members with conspiring to possess oxycodone with intent to distribute. If convicted, each suspect could face up to 20 years in prison.
As of Friday afternoon, nearly 20 of the suspects were in custody.
The case has been investigated by the Pasco County Sheriff's Office and the Drug Enforcement Administration. Authorities claim many of the suspects are addicts.
The suspects indicted are accused of obtaining the pills by doctor shopping and paying others for their pills.
According to DEA agents, oxycodone can be bought in Florida on the street for $10, but in Tennessee and Kentucky the pills can sell for $25-$35 a pill.
The Sheriff's Office apparently expects more arrests to be made.
Oxycodone crimes have become an increasing problem in Florida. Due to this, police have amped up their efforts to aggressively investigate those suspected of dealing in Oxycodone. Prosecutors take all Oxycodone offenses, including possession, trafficking and possession with intent to distribute quite seriously. In fact, the state of Florida has some of the toughest Oxycodone laws in the United States. It is a second-degree felony in Florida to sell, manufacture or deliver Oxycodone.
Oxycodone is a highly addictive painkiller, and its active ingredient is found in many prescription pain medications such as OxyContin and Roxicodone. Unfortunately, many people become addicted to Oxycodone after being prescribed the drug by a doctor following an accident or injury. After becoming addicted to these painkillers, addicts may facilitate their need for the drug by doctor shopping, forging prescriptions or finding alternative ways to acquire a large amount of pills. Because Oxycodone sells for a high amount on the streets, many people now sell the drug to support their own habit as well as to make a profit. These actions can result in severe state and federal charges carrying significant criminal consequences.
To give yourself the best chance at beating an Oxycodone charge, it is best to not admit any facts about your involvement with the drugs in question to police, prosecutors, court officials or friends and family members. There are many defense options available that may result in charges being dismissed or greatly reduced, so do not give up or plead guilty before fully exploring any and all possibilities. A plea agreement may also be available to you that can keep you out of prison and keep your record clean in exchange for drug rehabilitation, counseling or community service.
If you or someone you love has been charged with a felony drug crime in Pasco County, or anywhere in the state of Florida, contact the Florida Drug Crimes Defense Lawyers at Whittel & Melton online or call our Pasco County office at 727-847-2299 or reach us statewide and toll-free at 1-866-608-5LAW (5529).


text:
Q: QtQuick3D / Qt Design Studio GLTF import Material looks wrong i found this nice Car Seat for Blender, then exported the whole scene as GTLF2.0 and imported through Qt Design Studio as QML Asset.
I had some Timeline issue and I fixed this by hand, but one Material looks wrong as shown in the image.

I checked same GLTF2.0 File with a GLTF Onlineviewer Link and is looks equal to Blender.

Afterwards I checked with other GLTF Onlineviewer Link and here I have the same issue.

Questions:

*

*Does somebody know or give me hints which value could be the trouble
maker?

*Is not a lightning issue?

The Generated Qml File:
import QtQuick3D 1.15
import QtQuick 2.15
import QtQuick.Timeline 1.0

Node {
    id: rOOT

    property alias currentFrame: timeline0.currentFrame

    Node {
        id: light
        x: 407.625
        y: 590.386
        z: -100.545
        eulerRotation.x: -132.466
        eulerRotation.y: 72.7792
        eulerRotation.z: -169.256
        scale.x: 1
        scale.y: 1
        scale.z: 1

        PointLight {
            id: light_Orientation
            eulerRotation.x: -90
            color: "#ffffffff"
            brightness: 10000
            quadraticFade: 1
        }
    }

    Node {
        id: camera
        x: 524.195
        y: 308.297
        z: -485.887
        eulerRotation.x: -105.604
        eulerRotation.y: 44.346
        eulerRotation.z: -179.999

        PerspectiveCamera {
            id: camera_Orientation
            eulerRotation.x: -90
            clipNear: 0.1
            clipFar: 100
            fieldOfView: 39.5978
            fieldOfViewOrientation: Camera.Horizontal
        }
    }

    Node {
        id: empty_002

        Model {
            id: node01_Base
            y: 3.23133
            z: 21.8115
            source: "meshes/node01_Base.mesh"

            PrincipledMaterial {
                id: plastico_Negro_material
                baseColor: "#ff010101"
                metalness: 0
                roughness: 0.8
                normalMap: Texture {
                    source: "maps/0.png"

                }
                cullMode: Material.NoCulling
            }
            materials: [
                plastico_Negro_material
            ]
        }

        Model {
            id: node02_Base_rim
            y: 3.23133
            z: 21.8115
            eulerRotation.z: 180
            scale.x: 0.121224
            scale.y: 0.361819
            scale.z: 1.3171
            source: "meshes/node02_Base_rim.mesh"

            PrincipledMaterial {
                id: aluminio_galvanizado_material
                baseColor: "#ff383838"
                roughness: 0.5
                cullMode: Material.NoCulling
            }
            materials: [
                aluminio_galvanizado_material
            ]
        }

        Model {
            id: node03_Base_controls
            eulerRotation.x: 8.46812
            scale.x: 0.0367964
            scale.y: 0.271225
            scale.z: 0.271225
            source: "meshes/node03_Base_controls.mesh"

            PrincipledMaterial {
                id: aluminio_negro_material
                baseColor: "#ff080808"
                roughness: 0.5
                cullMode: Material.NoCulling
            }
            materials: [
                aluminio_galvanizado_material,
                aluminio_negro_material
            ]
        }

        Model {
            id: node04_Bottom_seat
            y: 32.6284
            tessellationMode: Model.NPatch
            receivesShadows: false
            isWireframeMode: false
            source: "meshes/node04_Bottom_seat.mesh"

            PrincipledMaterial {
                id: cUERO_NEGRO_material
                baseColor: "#ff534d30"
                metalness: 0
                roughness: 0.389465
                normalMap: Texture {
                    source: "maps/0.png"
                    tilingModeHorizontal: Texture.Repeat
                    tilingModeVertical: Texture.Repeat
                }
                cullMode: Material.NoCulling
            }
            materials: cUERO_NEGRO_material
        }

        Model {
            id: node05_Bottom_sides
            source: "meshes/node05_Bottom_sides.mesh"
            materials: cUERO_NEGRO_material
        }

        Model {
            id: node06_Back_seat
            y: -7.04048
            z: 6.17094
            eulerRotation.x: -1.59834
            source: "meshes/node06_Back_seat.mesh"
            materials: cUERO_NEGRO_material
        }

        Model {
            id: node07_Seat_back_sides
            source: "meshes/node07_Seat_back_sides.mesh"
            materials: cUERO_NEGRO_material
        }

        Model {
            id: node08_Upper_neck
            y: 222.546
            z: 199.645
            source: "meshes/node08_Upper_neck.mesh"
            materials: cUERO_NEGRO_material
        }

        Model {
            id: node09_Header
            isWireframeMode: true
            source: "meshes/node09_Header.mesh"
            materials: cUERO_NEGRO_material
        }
    }

    Timeline {
        id: timeline0
        startFrame: 0
        endFrame: 30
        currentFrame: 0
        enabled: true
        animations: [
            TimelineAnimation {
                duration: 30
                from: 0
                to: 30
                running: true
            }
        ]

        KeyframeGroup {
            target: empty_002
            property: "position"

            Keyframe {
                frame: 0
                value: Qt.vector3d(0, 0, 0)
            }
        }

        KeyframeGroup {
            target: empty_002
            property: "eulerRotation"

            Keyframe {
                frame: 0
                value: Qt.vector3d(0, 0, 0)
            }
            Keyframe {
                frame: 1
                value: Qt.vector3d(0, 6, 0)
            }
            Keyframe {
                frame: 2
                value: Qt.vector3d(0, 12, 0)
            }
            Keyframe {
                frame: 3
                value: Qt.vector3d(0, 18, 0)
            }
            Keyframe {
                frame: 4
                value: Qt.vector3d(0, 24, 0)
            }
            Keyframe {
                frame: 5
                value: Qt.vector3d(0, 30, 0)
            }
            Keyframe {
                frame: 291.667
                value: Qt.vector3d(0, 36, 0)
            }
            Keyframe {
                frame: 6
                value: Qt.vector3d(0, 42, 0)
            }
            Keyframe {
                frame: 7
                value: Qt.vector3d(0, 48, 0)
            }
            Keyframe {
                frame: 8
                value: Qt.vector3d(0, 54, 0)
            }
            Keyframe {
                frame: 9
                value: Qt.vector3d(0, 60, 0)
            }
            Keyframe {
                frame: 10
                value: Qt.vector3d(0, 66, 0)
            }
            Keyframe {
                frame: 11
                value: Qt.vector3d(0, 72, 0)
            }
            Keyframe {
                frame: 12
                value: Qt.vector3d(0, 78, 0)
            }
            Keyframe {
                frame: 13
                value: Qt.vector3d(0, 84, 0)
            }
            Keyframe {
                frame: 14
                value: Qt.vector3d(0, 90, 0)
            }
            Keyframe {
                frame: 15
                value: Qt.vector3d(0, 96, 0)
            }
            Keyframe {
                frame: 16
                value: Qt.vector3d(0, 102, 0)
            }
            Keyframe {
                frame: 17
                value: Qt.vector3d(0, 108, 0)
            }
            Keyframe {
                frame: 18
                value: Qt.vector3d(0, 114, 0)
            }
            Keyframe {
                frame: 19
                value: Qt.vector3d(0, 120, 0)
            }
            Keyframe {
                frame: 20
                value: Qt.vector3d(0, 126, 0)
            }
            Keyframe {
                frame: 21
                value: Qt.vector3d(0, 132, 0)
            }
            Keyframe {
                frame: 22
                value: Qt.vector3d(0, 138, 0)
            }
            Keyframe {
                frame: 23
                value: Qt.vector3d(0, 144, 0)
            }
            Keyframe {
                frame: 24
                value: Qt.vector3d(0, 150, 0)
            }
            Keyframe {
                frame: 25
                value: Qt.vector3d(0, 156, 0)
            }
            Keyframe {
                frame: 26
                value: Qt.vector3d(0, 162, 0)
            }
            Keyframe {
                frame: 27
                value: Qt.vector3d(0, 168, 0)
            }
            Keyframe {
                frame: 28
                value: Qt.vector3d(0, 174, 0)
            }
            Keyframe {
                frame: 29
                value: Qt.vector3d(0, 182, 0)
            }

            Keyframe {
                frame: 30
                value: Qt.vector3d(0, 190, 0)
            }

        }

        KeyframeGroup {
            target: empty_002
            property: "scale"

            Keyframe {
                frame: 0
                value: Qt.vector3d(1, 1, 1)
            }
        }
    }
}

A: The issue was in the PrincipledMaterial under Texture Mapping the properties
U Scale = 1
V Scale = 1
I changed both to 0, but nothing changed. I had to reload the Scene in the 3D Editor again.

The color difference is not a issue. I have to adapt a bit the light etc.
Here a extraction of the corrected PrincipledMaterial.
PrincipledMaterial {
                id: cUERO_NEGRO_material
                baseColor: "#ff534d30"
                metalness: 0
                roughness: 0.389465
                normalMap: Texture {
                    scaleV: 0
                    scaleU: 0
                    source: "maps/0.png"
                    tilingModeHorizontal: Texture.Repeat
                    tilingModeVertical: Texture.Repeat
                }
                cullMode: Material.NoCulling
            }
            materials: cUERO_NEGRO_material
        }

redpajama_set_name: RedPajamaStackExchange


text:
New powder-blue split recycle carts coming your way
Oct. 122010, 12:12 p.m. Sept. 52013
Berkeley Mayor Tom Bates shows off his new recycling cart outside his home. Photos: Tracey Taylor
A new era in recycling has begun in Berkeley with the introduction of light blue, split carts which carry paper waste in one side and bottles and cans in the other.
Mayor Tom Bates held a press conference outside his home on Ward Street this morning to unveil the new carts which replace the dark blue boxes currently in use. The new carts have already started to appear on curbsides and a total of 36,000 of them will be delivered to Berkeley residents in single family homes over the next month.
Seven new trucks designed to handle the new carts are also now in use — the trucks are also divided into two sections, the larger of which holds paper and cardboard materials.
Berkeley's recycling program has been managed by the Ecology Center since the early 1970s. Executive Director Martin Bourque said the new carts, which are modeled on those being used in Marin, offered three key advantages: reducing litter, reducing worker injuries and deterring poachers. "With the blue bins which are loaded manually, workers on some routes were having to lift up to 10,000 lbs a day," he said.
New trucks are handling the new carts
Accessing materials is more difficult than from an open box which might, Bourque said, put off potential poachers. The lid of the new cart is also printed with an advisory that the material contained in the cart is city property. Each compartment of the new carts has double the capacity of the old boxes. The closed carts also keep materials dry.
Mayor Bates said the carts represented a breakthrough. "They are one of the tools we need to be a zero-waste city."
As part of the city's Climate Action Plan, Berkeley's goal is to achieve zero waste by 2020. Waste sent to the landfill decreased 40% between 2000 and 2009, a reduction of more than 47,000 tons of waste.
Multi-family homes will receive two carts, one blue one for bottles and a brown one for paper. Residents who want to keep their old blue boxes may do so, said Bourque, but if they put them out on the curb they will be taken away on garbage collection day.
Berkeley's recycling program is regarded as one of the most well run in the country. However Bourque had a message for the city's residents: "We are not trying to make people recycle more, we want them to waste less."
Visit the Ecology Center for full information on the new carts and curbside recycling questions.
Update, 21.33: Reader Alicia helpfully points to the June 29 recommendation document to Berkeley City Council regarding the purchase agreement and loan agreement relating to the new split recycling carts which shows that the estimated cost of the new carts was $2,507,310.
Berkeley Climate Action PlanEcology CenterMayor Tom BatesRecycling
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redpajama_set_name: RedPajamaCommonCrawl


Blades stay sharp for clean, easy cutting.
High-carbon, stainless steel cutting surfaces remain sharp for years. Coated with black carbon and fluorine resin, which enables clean, quick and easy cuts without leaving residue.
This product was added to our catalog on Sunday 20 March, 2011.


I fell in love with Disney World in the 4th grade. It was the first trip I can remember taking to the magical world of Disney, and to my ten year old self, it was the most amazing place on earth. The night we left, I cried inconsolably, begging my parents to allow us to relocate to Florida. They didn't go for the idea. They did however start to take our family on yearly trips to Disney World.
My older brother went on to work at Disney World, and David and I visit several times a year. The magic never grows old, and we've learned a few insider tips on the way. Here's a list of our favorite magical Disney World moments that you can experience for yourself!
1) Walking along Main Street U.S.A. at night.
After the nightly Magic Kingdom fireworks show, there is a mass exodus out of the park, leaving Main Street U.S.A. virtually empty. We strongly encourage you to stick around to see it without the crowds. The walk is gorgeous (especially during the Holidays, when every inch is covered with tinsel and garlands), and full of old, turn of the century nostalgia.
2) Check out Disney's Board Walk.
The Board Walk resort has an actual, Atlantic City reproduction, of Seaside boardwalks of yesteryear. It comes alive at night with performers, golden lights on the water, and the sound of big band music.
3) Attend a castle lighting show at Christmas.
During the Holidays, Cinderella's castle is covered with hundreds of thousands of twinkle lights. At dusk, a show proceeds the moment they turn on all the lights. It truly is magic.
4) Do some Deluxe Resort hopping.
Go see the animals on the savannah at the Animal Kingdom Lodge, or the geyser explode at the Wilderness lodge. Spend some time listening to live music in the lobby of the Grand Floridian, or enjoy dole whip in a hammock at the Polynesian resort. Deluxe resorts are beautiful, and exploring them is an experience in itself.
5) Eat at one of Disney's Signature Restaurants.
Ambiance is king among Disney's fine dining. Whether its walls that softly change colors to mimic an African sunset at Jiko, or the floor to ceiling windows overlooking the Magic Kingdom at The California Grill, Disney signature restaurants are an immersive experience.
6) Ride Astro Orbiter at night.
Everything in Tomorrowland is lit up, and you have a great view of the castle changing colors. Bay Lake doesn't look bad from that vantage point either.
7) Taste your way around Epcot's World Showcase.
Some of our favorites are school bread in Norway, Creme Brûlée in France, humus and falafel in Morocco, and bratwurst in Germany. Happy eating!
8) Catch the rope dropping ceremony at Magic Kingdom.
Early in the morning, right before the park opens, there's music, performers, and the fab five arriving on train! Plus, you will get a great head start into the park ahead of most of the crowds.
9) Enjoy quirky Disney rides.
You can find magic in the most unlikely places at Disney World. Check out Captain EO, The Country Bear Jamboree, Carousel of Progress, and People Mover, for some less known, slightly eccentric attractions.
10) Enjoy one of Disney's incredible firework shows from a prime viewing spot.
Starbucks on Main Street, and we love to watch Illuminations from the bridge between England and France in World Showcase.


Thanks for the congratulations, Katie. We'll be getting married next year, for sure. I'm really looking forward to you returning the favor.
DUDE! OH DEAREST CUZ!! I love this! Didn't know you blog!!! LOVE LOVE LOVE IT!!! Oh SOOOo happy to have come to your neck of the woods....how are you these days? Would love to see all the wedding pictures!


text:
Sergej Makarov je jméno několika významných osobností:
 Sergej Makarov (oštěpař) (* 1973) – ruský oštěpař
 Sergej Makarov (lední hokejista) (* 1958) – sovětský a později ruský lední hokejista
 Sergej Makarov (fotbalista) (* 1996) – ruský fotbalista
 Sergej Makarov (volejbalista) (* 1980) – ruský volejbalista

Související články
 Makarov

redpajama_set_name: RedPajamaWikipedia


PermaFusion™ is a a revolution in topical delivery that maximizes effectiveness and minimizes risk. PermaFusion™ is a first of its kind, petrolatum-based, liquid-in-oil suspension technology that quite literally fuses water and oil. Active ingredients are delivered with maximum effectiveness, but our revolutionary technology permits the use of less ingredients, which minimizes or eliminates toxicity, irritation, and sensitization in clinical testing. Data from historical products, as well as in-vivo and in-vitro data, show the potential of Permafusion™ to deliver a new era of topical therapy.
Occlusive petrolatum carrier serves to enhance active ingredient delivery.
Low active ingredient load minimizes side effects while maximizing benefits.
Once in contact with the skin or nail surface, nanodroplets penetrate deep into the keratin.


This gluten-free loaf is absolutely delicious! It's most and full of flavour and makes a perfect treat any time of day.
Grease a loaf tin and line with baking paper. Preheat the oven to 170ºC.
Sift the flours, cinnamon, baking soda, baking powder and guar gum into a large bowl and combine really well with a whisk.
Add the brown sugar and chopped almonds and stir to combine.
Add the grated carrot, rice bran oil, eggs and milk and mix well.
Pour the mixture into the lined loaf tin and smooth the surface with the back of a spoon.
Bake in pre-heated oven for about 60 minutes or until golden and cooked. If the loaf starts to get too brown during cooking, cover with foil.
Remove from oven and cool in the loaf tin before turning out onto a wire rack to cool completely.
TIP! To test if the loaf is cooked, insert a skewer into the centre of the loaf and if it comes out clean the loaf is ready.
Delicious on its own or with a thin spread of softened butter.


text:
\section{Introduction}
\label{sec:intro}

5G services that are of mMTC type require support for a very high density of devices and low energy consumption~\cite{Wang2020}.
mMTC is the service category responsible for providing access to a large number of machine-type devices, such as those for sensing, monitoring, metering, and all other Internet-of-Things (IoT) related applications/services. Besides providing access to a large number of devices for this service, the other goal for these devices is to operate on an energy-efficient basis. As opposed to Ultra-Reliable Low-Latency Communications (URLLC), mMTC services are less stringent in terms of the network conditions needed for their successful operation, i.e., their service requirements are considerably more relaxed.

However, while the traffic for a single device underlying mMTC service is sporadic with only a few packets transmitted at a time, given the expected density of IoT devices,
multiplexing too many devices within the service area of a given cell
may put the cell
at risk of collapse by pushing its operation towards the capacity limits.
This imposes the need for careful resource and network planning in order to deliver the information of these devices in a timely manner.

There are some important research questions that arise from the problem of network planning for mMTC traffic:
\begin{itemize}
\item Given the traffic intensity of a user and its channel conditions, what is the amount of resources that are needed for  a (generally large number) of mMTC devices to send their data \emph{successfully} and \emph{timely} without collapsing the cellular network?
\item Given the finite network capacity, is it possible to serve a very large number of mMTC users within the cell without forcing the network to operate close to its critical capacity limits?
\end{itemize}

To answer these questions, the knowledge of \emph{traffic patterns} in the network is of paramount importance. With knowledge of traffic characteristics, resource planning can be conducted successfully. Also, given the limited network resources, deriving the traffic patterns can help the cellular operator to  determine the maximum number of mMTC users that can be served in a cell.

In this paper, we characterize completely the mMTC traffic pattern in the cell by deriving the distribution of inter-arrival times of the mMTC traffic at the base station and the distribution of the number of packets, given the statistics of traffic generation at individual mMTC users.
The analysis we provide in this paper relies on realistic assumptions, such as general distributions for the inter-generation times of data at mMTC users, and general distributions for the number of packets generated at once.
The results that we provide can help cellular network operators in planning the number of users that can be served (admission control), or in planning the network resources needed to provide service to a given number of mMTC users.
\textcolor{black}{The main message of this paper is that the traffic pattern that mMTC users generate and their number in the cell are far more decisive on the network traffic pattern than the channel conditions of mMTC users.}

Specifically, our main contributions are:
\begin{itemize}
\item We derive the distribution of inter-arrival times at the base station that holds for any distribution of the inter-generation times of the traffic at mMTC.
\item We also derive the distribution of the number of packets that arrive at once at the base station, for any distribution of the number of packets generated at once on mMTC devices within the cell.
\item Using extensive realistic simulations, conducted on real-life traces, \textcolor{black}{we show that the coefficient of variation of inter-arrival times at gNodeB is oblivious to the  channel conditions and traffic generation rates at mMTC users.}
\end{itemize}

The remainder of this paper is organized as follows. In Section~\ref{sec:model}, we introduce the system model, including the mMTC traffic. This is followed by the analysis for the distribution of inter-arrival times and the number of packets
in Section~\ref{sec:analysis}.
Then, we perform an evaluation on real traces in Section~\ref{sec:sims}.
In Section~\ref{sec:related}, we discuss some related work. Finally, Section~\ref{sec:conclusion} concludes the paper.

\section{Performance Modeling}
\label{sec:model}
In this section, we present the system model and traffic characteristics of mMTC users.

\subsection{System model}
We consider users within the coverage area of a \textcolor{black}{5G} macro base station (\textcolor{black}{gNodeB}) \textcolor{black}{in the} sub-6 GHz band. Due to the nature of mMTC traffic, we focus on the uplink.

The block resource allocation scheme in a time-slotted system (frames) is used in 5G, with \emph{physical resource blocks (PRB)} being the allocation unit~\cite{block}.
Within the frame, different blocks are assigned to different users. In general, the assignment will vary across frames.  As a result, scheduling needs to be performed along two dimensions, \emph{frequency} and \emph{time}.
The total number of available blocks for the users with mMTC traffic in the cell is $K$.

\textcolor{black}
The possibility of network slicing in 5G~\cite{p11} enables assigning \emph{dedicated} network resources to the same type of service, e.g., mMTC users.
Therefore, throughout this paper, we assume that all mMTC users belong to the same slice within the cell, and hence require the same service quality.}

In general, users will see different channel conditions in different frequencies (different blocks) even within the same frame. Therefore, they  will experience a different per-block \textcolor{black}{Signal-to-Interference-plus-Noise Ratio} (SINR). This  \textcolor{black}{is a function of the base station transmission power of the cell where users are, the transmission power of neighboring cells transmitting on the same frequency set (inter-cell interference), the background noise, and the corresponding channel gains~\cite{Fidan_TMC}.} Because of
the time-varying channel characteristics, the  per-block SINR changes from one frame to another even for the same block for every user. This changing \emph{per-block} SINR translates into a varying \emph{per-block rate} that is obtained using a modulation and coding scheme (MCS). In our system, we consider the MCS with $m$ possible values (the typical value of $m$ is 15)~\cite{3GPP_5G_NR}.

Further, for every user, we assume flat channels \textcolor{black}{(blocks)} in a frame, i.e., the per-block rate does not change during the frame, but it changes from one frame to another randomly.

Although, in general, different blocks will ``bring'' different per-block rates, for the sake of analytical tractability,
we make a simplifying assumption. Namely,
we assume that the gNodeB transmission power and channel characteristics of a user remain unchanged across all $K$ blocks in a frame. i.e., they undergo flat fading.
Consequently, our problem reduces to one-dimensional scheduling, in \emph{time}. Therefore, instead of deciding how many and which blocks to assign to every user, we use another parameter, defined as:
\begin{definition}
The ratio of the frame for which all the blocks are allocated to user $i$ is called \textbf{frame ratio}. It is denoted by $Y_i$ and can take values in the interval $[0,1]$.
\end{definition}

The blocks are assigned orthogonally during the frame so that no two  users receive them simultaneously.
\textcolor{black}{This is a reasonable simplification as frame ratio can be translated into the corresponding number of PRBs.}
Hence, the frame ratio will be the quantitative measure of resource allocation in this paper.

The previous assumptions enable us to model user's $i$ per-block rate as a discrete random variable, $R_{i}$, with values in $\left\{r_{1},r_{2},\ldots,r_{m}\right\}$, such that $r_{1}<r_{2}<\ldots<r_{m}$, \textcolor{black}{with a probability mass function (pmf) $p_{R_{i}}(x)$. The latter is a function of  user's $i$ SINR over time.}

\emph{Number of users:} There are $n$ users in the cell with different per-block rate distributions.

\subsection{mMTC traffic}
As already mentioned, we consider the communication on the uplink. The data from devices are sent to the base station that covers the cell where devices are located.

\emph{Traffic generation:} Data from every user to the base station are transmitted in packets. We assume that packet sizes, denoted as $\Theta$, are fixed~\cite{Ortiz}. This is reasonable as we are considering services in which the data are organized in small packets~\cite{Ortiz}. Data are generated with rate $\lambda$, which is called \emph{traffic generation rate}, and which represents the inverse of the average time between two consecutive transmissions of data from mMTC user. The mMTC traffic is sporadic, which means that it is characterized by small $\lambda$. The time interval between two consecutive data generation instants, which from now on we will be referring to as \emph{inter-generation time}, undergoes a general distribution.

\emph{Number of packets:} At the moment of generation, we assume that there are several packets that are transmitted. The random variable $N$ is used to denote the number of packets that are generated at a given time. We assume that $N(t)$ is a stationary stochastic process that is independent of the data rate (channel conditions) of the user.\textcolor{black}{\footnote{We remove the reference to $t$ for notational simplicity from now on.}} The maximum number of packets that an mMTC user can generate at once is $N_{max}$.

\textcolor{black}{Users operate independently from one another, i.e., the data generation instants and the number of packets at a specific user aren't in any way correlated with data generation instants and the number of packets of other users.}

\emph{Data rate:} The data rate a user experiences in a frame is $U_i=KY_iR_i$, where $Y_i$ is the frame ratio, $K$ is the total number of blocks, and $R_i$ is the random variable that denotes the per-block rate of user $i$ in a frame. In this paper, we assume that network resources are shared equally among the users, which implies a data rate of $U_i=\frac{KR_i}{n}$ for user $i$, where $n$ is the number of mMTC users in the cell.

As packet sizes are small, and not too many packets are transmitted at once, we assume that all the packets generated at once are transmitted with the same rate $U$.

\section{Analysis}
\label{sec:analysis}
In this section, we first derive the distribution of inter-arrival times of the traffic at gNodeB. Then, we derive the distribution of the number of packets arriving at gNodeB.

\subsection{Distribution of inter-arrival times}
First, we derive the distribution of inter-arrival times at the gNodeB from a single user. Then, we use this result to derive the distribution of inter-arrival times at gNode from all mMTC users in the cell.
\subsubsection{Single-user analysis}
\label{sec:analysis_single}
Let $t_i$ denote the time when the $i$'th amount of data are generated and  transmitted from a user to the base station. The number of packets in that batch is $N_i$. Note that $t_0=0$ and $N_0=0$. The time all these packets arrive at the base station is denoted by $t_i^{'}$, and is
\begin{equation}
t_i^{'}=t_i+\frac{N_i\Theta}{U_i}+t_{prop},
\label{eq:time_i}
\end {equation}
where $\frac{N_i\Theta}{U_i}$ represents the transmission delay, and $t_{prop}$ is the propagation delay from the user to the base station. Note that since we assume the mMTC traffic is sporadic and packet sizes are small, it is reasonable to assume that all the packets generated in an instant are to be transmitted within the same frame, whose duration is $10$\:ms~\cite{3GPP_5G_NR}.

The time instant at which the next batch of data from the same user arrives at the base station is
\begin{equation}
t_{i+1}^{'}=t_{i+1}+\frac{N_{i+1}\Theta}{U_{i+1}}+t_{prop}.
\label{eq:time_i+1}
\end{equation}
From Eqs.(\ref{eq:time_i}) and (\ref{eq:time_i+1}), we obtain the inter-arrival time between two data arrivals at gNodeB from the same user as
\begin{equation}
\Delta t^{'}=t_{i+1}^{'}-t_{i}^{'}=\Delta t + \Theta\left(\frac{N_{i+1}}{U_{i+1}}-\frac{N_i}{U_i}\right),
\label{eq:t_'}
\end{equation}
where $\Delta t= t_{i+1}-t_i$ is the period between two consecutive data generations at the mMTC user. Since we are assuming the equal-share policy for the network resources, the data rate of the user when generating the $i$'th batch of data is $U_i=\frac{KR_{i}}{n}$, and $U_{i+1}=\frac{KR_{i+1}}{n}$ when generating the $i+1$'th batch of data. Introducing the last assumptions into Eq.(\ref{eq:t_'}), we get
\begin{equation}
\Delta t^{'}=\Delta t + \frac{n\Theta}{K}\left(\frac{N_{i+1}}{R_{i+1}}-\frac{N_{i}}{R_{i}}\right).
\label{eq:t_'_updated}
\end{equation}
As the number of generated packets and user's per-block rate are independent from the moment when data are generated ($N(t)$ is a stationary stochastic process), $\Delta t$ is independent from the term $\frac{n\Theta}{K}\left(\frac{N_{i+1}}{R_{i+1}}-\frac{N_{i}}{R_{i}}\right)$
in Eq.(\ref{eq:t_'_updated}). Therefore, the probability density function (pdf) of $\Delta t^{'}$ can be written as~\cite{RossStochProc}
\begin{equation}
f_{\Delta t^{'}}(x)=f_{\Delta t}(x) * f_{\frac{n\Theta}{K}\left(\frac{N_{i+1}}{R_{i+1}}-\frac{N_i}{R_i}\right)}(x),
\label{eq:f}
\end{equation}
where * denotes the convolution operation~\cite{Oppenheim}.
\textcolor{black}{In Eq.(\ref{eq:f}), the second right-hand side term is a probability mass function (pmf). Namely, both $N$ and $R$ are discrete random variables (with values from finite discrete sets). Therefore, any combination of these two random variables with a constant ($\frac{n\Theta}{K}$) will take values from a finite discrete set. Hence, $\frac{n\Theta}{K}\left(\frac{N_{i+1}}{R_{i+1}}-\frac{N_i}{R_i}\right)$ is a discrete random variable, and $f_{\frac{n\Theta}{K}\left(\frac{N_{i+1}}{R_{i+1}}-\frac{N_i}{R_i}\right)}(x)$ is its pmf.\footnote{The inter-generation time of data at an mMTC user is a continuous random variable, the same as the inter-arrival time at gNodeB. Hence, the use of \emph{probability density function} (pdf) for the inter-arrival time.}}

Before proceeding with the derivation of the aforementioned pmf, let us show what happens to the pmf of a function when the latter is multiplied by a constant - in this case $\frac{n\Theta}{K}$.
\begin{lemma}
For random variable $X$ and constants $n, \Theta, K$, the following holds:
\begin{equation}
f_{\frac{n\Theta X}{K}}(x)=f_{X}\left(\frac{Kx}{n\Theta}\right).
\label{eq:lemma_1}
\end{equation}
\end{lemma}
\begin{proof}
$$f_{\frac{n\Theta X}{K}}(x)=\mathbb{P}\left(\frac{n\Theta X}{K}=x\right)=\mathbb{P}\left(X=\frac{Kx}{n\Theta}\right)=f_{X}\left(\frac{Kx}{n\Theta}\right)$$
\end{proof}
\noindent In order to proceed with the derivation, we need the following:
\begin{lemma}
The pmf of the difference between the ratio $\frac{N}{R}$ in two consecutive generations of data is
\begin{align}
f_{\frac{N_{i+1}}{R_{i+1}}-\frac{N_i}{R_i}}(x)&=\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\nonumber\\
&\cdot\delta\left(x-\frac{k}{r_i}+\frac{l}{r_j}\right),
\label{eq:NRx_8}
\end{align}
where $\delta(x)$ is the Dirac delta function~\cite{Oppenheim}, whose value is $1$ only at $x=0$, and $0$ otherwise.
\end{lemma}
\begin{proof}

The pmf of the random variable $\frac{N}{R}$ is
\begin{equation}
f_{\frac{N}{R}}(x)=\mathbb{P}\left(\frac{N}{R}=x\right)=\mathbb{P}\left(N=Rx\right)=f_{N}(Rx).
\end{equation}
For the latter, we have
\begin{eqnarray}
\mathbb{P}(N=Rx)&=\sum_{i=1}^{m}\mathbb{P}\left(N=Rx|R=r_i\right)\mathbb{P}\left(R=r_i\right)\nonumber\\
&=\sum_{i=1}^{m}\mathbb{P}\left(N=r_ix\right)p_R(r_i),
\label{eq:NRx}
\end{eqnarray}
where $p_R(r_i)$ denotes the probability mass function (pmf) of the per-block rate of a user. For the other term on the RHS of Eq.(\ref{eq:NRx}), we have
\begin{equation}
\mathbb{P}\left(N=r_ix\right)=p_{N}\left(r_ix\right)=\sum_{k=1}^{N_{max}}p_{N}(k)\delta\left(x-\frac{k}{r_i}\right).
\label{eq:NRx_1}
\end{equation}
Note that $\sum_{k=1}^{N_{max}}p_{N}(k)=1$ and $N_{max}$ is the maximum number of packets that can be generated at once.
Substituting Eq.(\ref{eq:NRx_1}) into Eq.(\ref{eq:NRx}), we obtain
\begin{equation}
f_{\frac{N}{R}}(x)=\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}p_{N}(k)p_{R}(r_i)\delta\left(x-\frac{k}{r_i}\right).
\label{eq:NRx_2}
\end{equation}
Next, we need to look at the difference of two i.i.d. random variables
$\frac{N}{R}-\frac{N}{R}=\frac{N}{R}+\left(-\frac{N}{R}\right)$.\footnote{Note that this is the difference of two \emph{random} variables, and not the difference of two ordinary variables. Hence, it is not $0$ in general.} So, we need the pmf of the random variable $-\frac{N}{R}$. It is
\begin{equation}
f_{\hspace{-3pt}-\frac{N}{R}}(x)\hspace{-3pt}=\mathbb{P}\left(-\frac{N}{R}=x\right)=\mathbb{P}\left(\frac{N}{R}=-x\right)= f_{\frac{N}{R}}(\hspace{-3pt}-x),
\end{equation}
which replaced into Eq.(\ref{eq:NRx_2}) yields
\begin{equation}
f_{-\frac{N}{R}}(x)=\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}p_{N}(k)p_{R}(r_i)\delta\left(-x-\frac{k}{r_i}\right).
\label{eq:NRx3}
\end{equation}
In line with Eqs.(\ref{eq:t_'}) and (\ref{eq:f}), the following holds
\begin{equation}
f_{\frac{N_{i+1}}{R_{i+1}}-\frac{N_i}{R_i}}(x)=f_{\frac{N}{R}}(x)*f_{\frac{N}{R}}(-x).
\label{eq:NRx_4}
\end{equation}
Substituting Eqs.(\ref{eq:NRx_2}) and (\ref{eq:NRx3}) into Eq.(\ref{eq:NRx_4}) results in
\begin{eqnarray}
f_{\frac{N_{i+1}}{R_{i+1}}-\frac{N_i}{R_i}}(x)=\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}p_N(k)p_R(r_i)\delta\left(x-\frac{k}{r_i}\right)\nonumber\\
*\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(l)p_R(r_j)\delta\left(-x-\frac{l}{r_j}\right).
\label{eq:NRx_5}
\end{eqnarray}
Note that the convolution operation is characterized by the distributive property~\cite{Oppenheim}, i.e., $f(x)*\sum_{i} g_i(x)=\sum_i f(x)*g_i(x)$. Therefore, Eq.(\ref{eq:NRx_5}) transforms into
\begin{align}
f_{\frac{N_{i+1}}{R_{i+1}}-\frac{N_i}{R_i}}(x)&=
\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\nonumber\\
&\cdot\delta\left(x-\frac{k}{r_i}\right)*\delta\left(-x-\frac{l}{r_j}\right).
\label{eq:NRx_6}
\end{align}
For the convolution of two $\delta$ functions from Eq.(\ref{eq:NRx_6}), we have
\begin{align}
&\delta\left(x-\frac{k}{r_i}\right)*\delta\left(-x-\frac{l}{r_j}\right)=\delta\left(-x-\frac{l}{r_j}\right)*\delta\left(x-\frac{k}{r_i}\right)\nonumber\\
&=\delta\left(-\left(x-\frac{k}{r_i}\right)-\frac{l}{r_j}\right)=\delta\left(-x+\frac{k}{r_i}-\frac{l}{r_j}\right).
\label{eq:aux_0}
\end{align}
From the definition of the Dirac delta function,
$\delta\left(-x+\frac{k}{r_i}-\frac{l}{r_j}\right)=1$
only when
$-x+\frac{k}{r_i}-\frac{l}{r_j}=0$,
which yields
$x=\frac{k}{r_i}-\frac{l}{r_j}$. Therefore,
\begin{equation}
\delta\left(-x+\frac{k}{r_i}-\frac{l}{r_j}\right)=\delta\left(x-\frac{k}{r_i}+\frac{l}{r_j}\right).
\label{eq:NRx_7}
\end{equation}
Substituting Eq.(\ref{eq:NRx_7}) into Eq.(\ref{eq:aux_0}), and the latter into Eq.(\ref{eq:NRx_6}), we obtain Eq.(\ref{eq:NRx_8}).
\end{proof}

\noindent Using the results from Lemma~1 and Lemma~2, we obtain:
\begin{theorem}
The pdf of inter-arrival times between two consecutive data batches at gNodeB from a single mMTC user is
\begin{align}
f_{\Delta t^{'}}(x)&=\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\nonumber\\
&\cdot f_{\Delta t}\left(x-\frac{n\Theta}{K}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)\right).
\label{eq:NRx_13}
\end{align}
\end{theorem}
\begin{proof}
From Eq.(\ref{eq:lemma_1}), we have
\begin{equation}
f_{\frac{n\Theta}{K}\left(\frac{N_{i+1}}{R_{i+1}}-\frac{N_i}{R_i}\right)}(x) =f_{\frac{N_{i+1}}{R_{i+1}}-\frac{N_i}{R_i}}\left(\frac{Kx}{n\Theta}\right),
\label{eq:cx}
\end{equation}
which from Lemma~2 yields
\begin{align}
f_{\frac{n\Theta}{K}\left(\frac{N_{i+1}}{R_{i+1}}-\frac{N_i}{R_i}\right)}(x)&\hspace{-3pt}=\hspace{-3pt}\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}\hspace{-3pt}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\nonumber\\
&\cdot\delta\left(\frac{Kx}{n\Theta}-\frac{k}{r_i}+\frac{l}{r_j}\right).
\label{eq:NRx_9}
\end{align}
Following a similar reasoning for the argument of $\delta$ as before, we have
\begin{equation}
\delta\left(\frac{Kx}{n\Theta}-\frac{k}{r_i}+\frac{l}{r_j}\right)=\delta\left(x-\frac{n\Theta}{K}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)\right).
\label{eq:NRx_10}
\end{equation}
Substituting Eq.(\ref{eq:NRx_10}) into Eq.(\ref{eq:NRx_9}), we obtain
\begin{align}
f_{\frac{n\Theta}{K}\left(\frac{N_{i+1}}{R_{i+1}}-\frac{N_i}{R_i}\right)}(x)&=\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\nonumber\\
&\cdot\delta\left(x-\frac{n\Theta}{K}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)\right).
\label{eq:NRx_11}
\end{align}
As for the convolution operation the Dirac delta function is the unit element and it holds that $f(x)*\delta(x-x_0)=f(x-x_0)$, we have
\begin{equation}
f_{\Delta t}(x)*\delta\left(x-\frac{n\Theta}{K}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)\right)=f_{\Delta t}\left(x-\frac{n\Theta}{K}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)\right)
\label{eq:NRx_12}
\end{equation}
Substituting Eq.(\ref{eq:NRx_11}) into Eq.(\ref{eq:f}), and
due to the distributive property of convolution and Eq.(\ref{eq:NRx_12}), we obtain Eq.(\ref{eq:NRx_13}).
\end{proof}
Eq.(\ref{eq:NRx_13}) represents the pdf of inter-arrival times of data at gNodeB from a specific user. Next, we derive the distribution of inter-arrival times to gNodeB from all the mMTC users in the cell.

\subsubsection{Multi-user analysis}
\label{sec:analysis_multiple}
Let $T_j$ denote the inter-arrival time of data at gNodeB from mMTC user $j\in\{1,\ldots,n\}$. When the process starts at $t=0$ the first data arrival is described by
\begin{equation}
    \min \{T_1,\ldots,T_n\}.
\end{equation}
After the first arrival and when the equilibrium is established,
the stochastic arrival process at the gNodeB can be captured by (subscript $m$ specifies the multi-user case)
\begin{equation}
    \Delta t_m^{'}=\min \{T_1^{(e)},\ldots,T_n^{(e)}\},
\end{equation}
where $T_{j}^{(e)}$ denotes the \emph{excess distribution}\footnote{This is the time until the next arrival of data from user $j$ occurs.} of the inter-arrival time for packets from user $j$ at gNodeB.

The cumulative distribution function (CDF) of the inter-arrival time at the base station is
\begin{equation}
F_{\Delta t_{m}^{'}}(x)=\mathbb{P}(\Delta t_{m}^{'}\leq x)=\mathbb{P}\left(\min \{T_1^{(e)},\ldots,T_n^{(e)}\}\leq x\right).
\label{eq:TB}
\end{equation}
If the minimum of a number of random variables is smaller than or equal to $x$, then each of these variables has to be smaller than or equal to $x$, leading to
\begin{equation}
\mathbb{P}(\min \{T_1^{(e)},\ldots,T_n^{(e)}\}\leq x)=\mathbb{P}(T_1^{(e)}\leq x, \ldots, T_n^{(e)}\leq x).
\end{equation}
As the processes of data generation at the mMTC users are mutually independent, so are the inter-arrival times of their data at gNodeB. Hence, we have
\begin{equation}
\mathbb{P}(T_1^{(e)}\leq x, \ldots, T_n^{(e)}\leq x)=\prod_{j=1}^{n}\mathbb{P}(T_j^{(e)}\leq x)=\prod_{j=1}^{n}F_{T_j^{(e)}}(x)   \label{eq:min}
\end{equation}
From the theory of \emph{renewal processes}\cite{RossStochProc}, it is known that the pdf of the excess distribution is given b
\begin{equation}
f_{T^{(e)}}(x)=\frac{1-F_{\Delta t^{'}}(x)}{\mathbb{E}[\Delta t ^{'}]}.
\label{eq:Te}
\end{equation}
We have already shown in Section~\ref{sec:analysis_single} that the distribution of inter-arrival times from a user to gNodeB is not the same as the distribution of  inter-generation times for the data at the mMTC user. However, \emph{what happens with the average inter-arrival rate of data from an mMTC user to gNodeB}? The answer to this question is as follows.
\begin{lemma}
The inter-arrival rate of traffic at gNodeB from a specific user is identical to the inter-generation rate of the traffic of that same mMTC user, i.e.,
\begin{equation}
\lambda^{'}=\lambda.
\label{eq:lambda=lambda'}
\end{equation}
\end{lemma}
\begin{proof}
The traffic generation rate at the user is
\begin{equation}
\lambda=\frac{1}{\mathbb{E}[t_i-t_{i-1}]},
\label{eq:lambda}
\end{equation}
i.e., the inverse
of the mean inter-generation times.
On the other hand, the rate at which the traffic from the same user arrives at the base station is
\begin{equation}
\lambda^{'}=\frac{1}{\mathbb{E}[t_i^{'}-t_{i-1}^{'}]}.
\label{eq:lambda'}
\end{equation}
Substituting Eqs.(\ref{eq:time_i}) and (\ref{eq:time_i+1}) into Eq.(\ref{eq:lambda'}), we obtain
\begin{equation}
\lambda^{'}=\frac{1}{\mathbb{E}\left[\Delta t_i+\frac{n\Theta}{K}\left(\frac{N_{i}}{R(t_i)}-\frac{N_{i-1}}{R(t_{i-1})}\right)\right]},
\label{eq:lambda'_2}
\end{equation}
where $\Delta t_i=t_i-t_{i-1}$ is the inter-generation time between the $i-1$'th and $i$'th data generation at the user.

Let us now take a closer look at the term $\mathbb{E}\left[\frac{N_{i}}{R(t_i)}-\frac{N_{i-1}}{R(t_{i-1})}\right]$. Since $N(t)$ is a stationary stochastic process independent of $R(t)$, and $R(t)$ is independent from one frame to another,
we have
\begin{align}
\mathbb{E}\left[\frac{N_{i}}{R(t_i)}\right]&=\mathbb{E}\left[N_i\right]\mathbb{E}\left[\frac{1}{R(t_i)}\right]=\mathbb{E}\left[N_{i-1}\right]\mathbb{E}\left[\frac{1}{R(t_{i-1})}\right]\nonumber\\
=\mathbb{E}\left[\frac{N_{i-1}}{R(t_{i-1)}}\right],
\end{align}
which after being substituted into Eq.(\ref{eq:lambda'_2}) implies
\begin{equation}
\lambda^{'}=\frac{1}{\mathbb{E}[\Delta t_i]}=\lambda.
\end{equation}
\end{proof}
Next, for the CDF of inter-arrival times at gNodeB from all the mMTC users within the cell, we have:
\begin{theorem}
The CDF of inter-arrival times at gNodeB from $n$ mMTC users in the cell is
\begin{equation}
F_{\Delta t_m^{'}}(x)=\prod_{i=1}^{n}\lambda_i\int_{0}^{x}\left(1-F_{\Delta t_i^{'}}(w)\right)dw,
\label{eq:theorem_5}
\end{equation}
where $F_{\Delta t_{i}^{'}}(w)$ is the CDF of inter-arrival times of data from user $i$ to gNodeB.
\end{theorem}
\begin{proof}
From Eq.(\ref{eq:Te}) and Eq.(\ref{eq:lambda=lambda'}), we get
\begin{equation}
f_{T_{i}^{(e)}}(x
=\lambda_i\left(1-F_{\Delta t_{i}^{'}}(x)\right).
\label{eq:Te_1}
\end{equation}
The CDF of the excess distribution is
\begin{equation}
F_{T_{i}^{(e)}}(x)=\int_{0}^{x}f_{T_{i}^{(e)}}(w)dw=\lambda_i\int_{0}^{x}(1-F_{\Delta t_{i}^{'}}(w))dw,
\label{eq:FTe}
\end{equation}
which substituted into Eq.(\ref{eq:min}) yields Eq.(\ref{eq:theorem_5}).
\end{proof}
Note that
\begin{equation}
F_{\Delta t^{'}}(w)=\int_{0}^{w}f_{\Delta t^{'}}(y)dy
\label{eq:FTX_aux}
\end{equation}
is the CDF of inter-arrival times of data from an mMTC user to gNodeB.

\subsection{Distribution of the number of packets}
Having derived the distribution of inter-arrival times of data at gNodeB, we proceed with looking into the distribution of data arriving at once during these arrival instants.

Determining the distribution of the number of packets at a time instant reduces to deciding which user the received data belong to. Therefore, the received data belong to user 1 (drawn from distribution $N_1$) with probability $p_1$, user 2 (drawn from distribution $N_2$), \ldots, user $n$ (drawn from distribution $N_n$). Hence, the pmf of the number of packets arriving at once at gNodeB is
\begin{equation}
p_N(x)=\sum_{i=1}^{n}p_ip_{N_i}(x).
\label{eq:pdf_N}
\end{equation}

The next step is deriving the probabilities $p_i, \forall i\in\{1,\ldots,n\}$. As mentioned previously, $p_i$ denotes the probability that the next arrival of data belongs to user $i$, i.e., the excess distribution of the inter-arrival time of user $i$ has the lowest value. According to this definition, for $p_i$ we have
\begin{equation}
p_i=\mathbb{P}\left(T_i^{(e)}\leq \min\{T_1^{(e)},\ldots,T_n^{(e)}\}\right),
\label{eq:pi_0}
\end{equation}
where under the \emph{min} operator there is no $T_{i}^{(e)}$ term. The following holds:
\begin{theorem}
The pmf of the number of packets arriving at once at gNodeB is given by
\begin{equation}
p_{N}(x)=\sum_{i=1}^{n}p_{N_i}(x)\prod_{j=1, j\neq i}^{n}\int_{0}^{\infty}F_{T_{i}^{(e)}}(x)f_{T_j^{(e)}}(x)dx.
\label{eq:N_final}
\end{equation}
\end{theorem}
\begin{proof}
Eq.(\ref{eq:pi_0}) is equivalent to
\begin{equation}
p_i=\mathbb{P}\left(T_i^{(e)}\leq T_1^{(e)},\ldots, T_i^{(e)}\leq T_n^{(e)}\right),
\end{equation}
and further, as inter-arrival times from individual mMTC users to gNodeB are mutually independent and hence the independence of their excess distributions, we have
\begin{equation}
p_i=\prod_{j=1, j\neq i}^{n}\mathbb{P}\left(T_{i}^{(e)}\leq T_{j}^{(e)}\right). \label{eq:pi}
\end{equation}
Next, we need to derive $\mathbb{P}\left(T_{i}^{(e)}\leq T_{j}^{(e)}\right)$. We do this by conditioning on the value of $T_{j}^{(e)}$. This yields
\begin{equation*}
\mathbb{P}\left(T_{i}^{(e)}\leq T_{j}^{(e)}\right)=\int_{0}^{\infty}\mathbb{P}\left(T_{i}^{(e)}\leq T_{j}^{(e)}|T_{j}^{(e)}=x\right)f_{T_j^{(e)}}(x)dx,
\end{equation*}
\begin{equation}
=\int_{0}^{\infty}\mathbb{P}\left(T_{i}^{(e)}\leq x\right)f_{T_j^{(e)}}(x)dx=\int_{0}^{\infty}F_{T_{i}^{(e)}}(x)f_{T_j^{(e)}}(x)dx \label{eq:P_excess},
\end{equation}
where $f_{T_j^{(e)}}(x)$ is defined by Eq.(\ref{eq:Te}).

Finally, substituting Eq.(\ref{eq:P_excess}) into Eq.(\ref{eq:pi}), and the latter into Eq.(\ref{eq:pdf_N}), we obtain Eq.(\ref{eq:N_final}).
\end{proof}

\subsection{Examples of specific distributions}
\label{sec:specific}

In this section, we look at the distribution of inter-arrival times and the number of packets for four special cases of distributions of inter-generation times at mMTC users: deterministic, uniform, exponential and Pareto distributed.
The first two are characterized by low variances (increasing failure rates), the third is memoryless, and Pareto distributions are characterized by heavy tails (decreasing failure rates). To ease the presentation, we assume that all mMTC users have i.i.d. traffic generation patterns.

\subsubsection{Deterministic inter-generation times}
In this case, the pdf of the inter-generation time at the mMTC user is given by
\begin{equation}
f_{\Delta t}(x)=\delta\left(x-\frac{1}{\lambda}\right),
\label{eq:deterministic}
\end{equation}
where $\frac{1}{\lambda}$ is the (fixed) time between two consecutive generations of data. Substituting Eq.(\ref{eq:deterministic}) into Eq.(\ref{eq:NRx_13}), we get
\begin{eqnarray}
f_{\Delta t^{'}}(x)=\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\cdot\nonumber\\
\delta\left(x-\frac{n\Theta}{K}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)-\frac{1}{\lambda}\right),
\label{eq:deterministic_1}
\end{eqnarray}
and the latter into Eq.(\ref{eq:FTX_aux}), we obtain
\begin{eqnarray}
F_{\Delta t^{'}}(x)=\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\nonumber\\
\cdot\int_{0}^{x}\delta\left(w-\frac{n\Theta}{K}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)-\frac{1}{\lambda}\right)dw.
\label{eq:deterministic_2}
\end{eqnarray}
This yields
\begin{eqnarray}
F_{\Delta t^{'}}(x)=\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\nonumber\\
\cdot u\left(x-\frac{n\Theta}{K}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)-\frac{1}{\lambda}\right),
\label{eq:deterministic_2_2}
\end{eqnarray}
where $u(x)$ denotes the Heaviside function (unit step function)~\cite{Oppenheim}, whose value is $1$ for $x\geq 0$. Otherwise, its value is $0$. In our case,
as long as $x\geq \frac{n\Theta}{K}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)+\frac{1}{\lambda}$, the value of the $u$-function in Eq.(\ref{eq:deterministic_2_2}) is $1$. Otherwise, it is $0$.

For mMTC users with homogeneous traffic patterns, Eq.(\ref{eq:theorem_5}) reduces to
\begin{equation}
F_{\Delta t^{'}_{m}}(x)=\lambda^{n}\left(\int_{0}^{x}\left(1-F_{\Delta t^{'}}(w)\right)dw\right)^{n}.
\label{eq:homogeneous_iid}
\end{equation}

Substituting Eq.(\ref{eq:deterministic_2_2}) into Eq.(\ref{eq:homogeneous_iid}), and computing numerically the integral, we obtain the CDF of the inter-arrival time of data at gNodeB for this case.

As far as the pmf of the number of packets arriving at gNodeB is concerned, for homogeneous mMTC traffic, Eq.(\ref{eq:N_final}) reduces to
\begin{equation}
p_N(x)=\lambda^{n-1}\sum_{i=1}^{n}p_{N_i}(x)\left(\int_{0}^{\infty}F_{T^{(e)}}(x)\left(1-F_{\Delta t^{'}}(x)\right)dx\right)^{n-1}
\label{eq:deterministic_final_packets}
\end{equation}
where $f_{T^{(e)}}(x)$ in Eq.(\ref{eq:N_final}) was replaced by Eq.(\ref{eq:Te}), i.e.,
\begin{equation}
f_{T^{(e)}}(x)=\frac{1-F_{\Delta t^{'}}(x)}{\mathbb{E}[\Delta t^{'}]}=\lambda\left(1-F_{\Delta t^{'}}(x)\right).
\label{eq:excess_iid}
\end{equation}
The second term under the sum in Eq.(\ref{eq:deterministic_final_packets}) is not a function of $i$. Hence, it goes out of the sum. Also, as we are dealing with i.i.d. traffic patterns,
\begin{equation}
\sum_{i=1}^{n}p_{N_i}(x)=n p_{N}(x).
\end{equation}
Now, Eq.(\ref{eq:deterministic_final_packets}) transforms into
\begin{equation}
p_{N}(x)=n\lambda^{n-1}p_{N}(x)\left(\int_{0}^{\infty}F_{T^{(e)}}(x)\left(1-F_{\Delta t^{'}}(x)\right)dx\right)^{n-1}.
\label{eq:deterministic_final_packets_0}
\end{equation}

In Eq.(\ref{eq:deterministic_final_packets_0}), $F_{\Delta t^{'}}(x)$ is given by Eq.(\ref{eq:deterministic_2_2}), and $F_{T^{(e)}}(x)$ is given by Eq.(\ref{eq:FTe}).

\subsubsection{Uniform inter-generation times}
The pdf of the inter-generation time of data in this case is
\begin{equation}
f_{\Delta t}(x)=\frac{1}{b-a}\left(u(x-a)-u(x-b)\right),
\label{eq:uniform}
\end{equation}
where $a$ and $b$ are the minimum and maximum possible inter-generation times at the mMTC user, and $u(x)$ is the unit step function introduced previously.
Note that the average inter-generation time in this case is $\frac{a+b}{2}=\frac{1}{\lambda}$.

Substituting Eq.(\ref{eq:uniform}) into Eq.(\ref{eq:NRx_13}), and the latter into Eq.(\ref{eq:FTX_aux}), after rearranging we obtain
\begin{align}
F_{\Delta t^{'}}(x)&=\frac{1}{b-a}\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\cdot\nonumber\\
&\left(\int_{0}^{x}u\left(w-\frac{n\Theta}{K}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)-a\right)dw\right.\nonumber\\
&\left.-\int_{0}^{x}u\left(w-\frac{n\Theta}{K}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)-b\right)dw\right)
\label{eq:uniform_2}
\end{align}
Finally, replacing Eq.(\ref{eq:uniform_2}) into Eq.(\ref{eq:homogeneous_iid}), and computing numerically the corresponding integrals, we obtain the CDF of the inter-arrival time distribution at gNodeB for this case.

The pmf of the number of packets arriving at gNodeB, for this case, is obtained from
\begin{equation}
p_{N}(x)=n\lambda^{n-1}p_{N}(x)\left(\int_{0}^{\infty}F_{T^{(e)}}(x)\left(1-F_{\Delta t^{'}}(x)\right)dx\right)^{n-1}
\end{equation}
with $F_{\Delta t^{'}}(x)$ given by Eq.(\ref{eq:uniform_2}), and $F_{T^{(e)}}(x)$ given by Eq.(\ref{eq:FTe}).

\subsubsection{Exponential inter-generation times}
The pdf of the inter-generation time of data in this case is
\begin{equation}
f_{\Delta t}(x)=\lambda e^{-\lambda x},
\label{eq:exponential_0}
\end{equation}
where $\frac{1}{\lambda}$ is the average inter-generation time. Substituting Eq.(\ref{eq:exponential_0}) into Eq.(\ref{eq:NRx_13}), after rearranging we obtain
\begin{align}
f_{\Delta t^{'}}(x)&=\lambda e^{-\lambda x}\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\nonumber\\
&\cdot e^{\frac{\lambda n K}{\Theta}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)}.
\label{eq:exponential}
\end{align}
Replacing Eq.(\ref{eq:exponential}) into Eq.(\ref{eq:FTX_aux}), after solving the corresponding integral and rearranging, we obtain
\vspace{-6pt}

\footnotesize
\begin{align}
F_{\Delta t^{'}}(w)&=\left(1-e^{-\lambda w}\right)\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\nonumber\\
&\cdot e^{\frac{\lambda n K}{\Theta}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)}.
\label{eq:exponential_2}
\end{align}
\normalsize
Finally, substituting Eq.(\ref{eq:exponential_2}) into Eq.(\ref{eq:homogeneous_iid}), solving the corresponding integral, and rearranging, we get
\begin{equation}
F_{\Delta t_{m}}^{'}(x)=\lambda^{n} \left[(1-A)x+\frac{A}{\lambda}\left(1-e^{-\lambda x}\right)\right]^{n},
\label{eq:exponential_3}
\end{equation}
where $$A=\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)e^{\frac{\lambda n K}{\Theta}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)}$$
As can be observed from Eq.(\ref{eq:exponential_3}), in this scenario we can obtain a closed-form expression for the CDF of inter-arrival time at  gNodeB.

The pmf of the number of packets arriving at gNodeB is
\begin{equation}
p_{N}(x)=n\lambda^{n-1}p_{N}(x)\left(\int_{0}^{\infty}F_{T^{(e)}}(x)\left(1-F_{\Delta t^{'}}(x)\right)dx\right)^{n-1}
\end{equation}
with $F_{\Delta t^{'}}(x)$ given by Eq.(\ref{eq:exponential_2}), and $F_{T^{(e)}}(x)$ given by Eq.(\ref{eq:FTe}).

\subsubsection{Pareto-distributed inter-generation times}
The pdf of the inter-generation time in this case is
\begin{equation}
f_{\Delta t}(x)=\frac{\alpha a^{\alpha}}{x^{\alpha+1}},
\label{eq:Pareto_0}
\end{equation}
where $a$ is the minimum possible inter-generation time, and $\alpha$ is the shape parameter of the Pareto distribution. We assume that $\alpha>1$. The average of Pareto distribution is $\frac{1}{\lambda}=\frac{a\alpha}{\alpha-1}$, resulting in $a=\frac{\alpha-1}{\alpha\lambda}$.

The pdf of the inter-arrival time at gNodeB is (after replacing Eq.(\ref{eq:Pareto_0}) into Eq.(\ref{eq:NRx_13}))
\begin{align}
f_{\Delta t^{'}}(x)&=\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\nonumber\\
&\cdot\frac{\alpha a^\alpha}{\left(x-\frac{n\Theta}{K}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)\right)^{\alpha+1}}.
\label{eq:Pareto_2}
\end{align}
Substituting Eq.(\ref{eq:Pareto_2}) into Eq.(\ref{eq:FTX_aux}), after solving the integral and rearranging, we obtain
\begin{eqnarray}
F_{\Delta t^{'}}(x)=a^{\alpha}\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\nonumber\\
\cdot\left(\hspace{-3pt}-\hspace{-3pt}\left(x\hspace{-3pt}-\hspace{-3pt}\frac{n\Theta}{K}\hspace{-3pt}\left(\frac{k}{r_i}\hspace{-3pt}-\hspace{-3pt}\frac{l}{r_j}\right)\right)^{\hspace{-3pt}-\alpha}\hspace{-3pt}\hspace{-3pt}+\hspace{-3pt}\left(\hspace{-3pt}-\hspace{-3pt}\frac{n\Theta}{K}\hspace{-3pt}\left(\frac{k}{r_i}\hspace{-3pt}-\hspace{-3pt}\frac{l}{r_j}\right)\right)^{\hspace{-3pt}-\alpha}\right)
\label{eq:Pareto}
\end{eqnarray}
Finally, substituting Eq.(\ref{eq:Pareto}) into Eq.(\ref{eq:homogeneous_iid}), solving the integral and rearranging the terms, we obtain the CDF of the inter-arrival time of data packets at gNodeB as Eq.(\ref{eq:Pareto_3}), where $B=\frac{n\Theta}{K}\left(\frac{k}{r_i}-\frac{l}{r_j}\right)$. In this scenario as well, we can find the distribution of the inter-arrival time at gNodeB in closed form.
\begin{figure*}[b]
\begin{equation}
F_{\Delta t_{m}^{'}}(x)=\lambda^{n}\left[x+a^{\alpha}\sum_{i=1}^{m}\sum_{k=1}^{N_{max}}\sum_{j=1}^{m}\sum_{l=1}^{N_{max}}p_N(k)p_R(r_i)p_N(l)p_R(r_j)\left(\frac{\left(x-B\right)^{1-\alpha}}{1-\alpha}-\frac{\left(-B\right)^{1-\alpha}}{1-\alpha}-\left(-B\right)^{-\alpha}x\right)\right]^{n}
\label{eq:Pareto_3}
\end{equation}
\end{figure*}

The pmf of the number of packets arriving at gNodeB is
\begin{equation}
p_{N}(x)=n\lambda^{n-1}p_{N}(x)\left(\int_{0}^{\infty}F_{T^{(e)}}(x)\left(1-F_{\Delta t^{'}}(x)\right)dx\right)^{n-1},
\end{equation}
with $F_{\Delta t^{'}}(x)$ given by Eq.(\ref{eq:Pareto}), and $F_{T^{(e)}}(x)$ given by Eq.(\ref{eq:FTe}).

\section{Performance Evaluation}
\label{sec:sims}
We first describe the simulation setup. Then, we focus on the impact some of the parameters have on the base station traffic patterns.

\subsection{Simulation setup}
As input parameters for the signal quality, we have used a 5G trace with data measured in the Republic of Ireland.
These traces can be found in~\cite{trace_5G_link}, with a detailed description in~\cite{trace_paper_5G}.
The parameter of interest from the trace is CQI (Channel Quality Indicator) with $15$ levels, which serves to determine the per-block rate of a user in a frame. These measurements were conducted for one user, but on different days, for different applications, and when the user is static and driving  around. To mimic the predominantly static nature of mMTC devices, we picked 6 ``basic''\footnote{We call them basic users as we will replicate these users to create more users in the corresponding scenarios, as needed.} users that are static, and assume they all are in the same cell.
Based on the frequency of occurrence of a per-block rate for every user, we obtained the corresponding per-block rate probabilities in Table~\ref{Table-R_2}.

The packet size in all the scenarios is $5$\:kbits.\footnote{We tried other values for  packet sizes with similar conclusions drawn.} Unless stated otherwise, mMTC users are placed uniformly in a distance $(0.1,3)$\:km from the gNodeB. The propagation delay for every user is found as $t_{prop}=\frac{d}{c}$, where $d$ is the distance of the user from gNodeB, and $c=3\cdot10^8$\:m/s.

\textcolor{black}{The development of network slicing in 5G~\cite{p11} has allowed operators to split users into groups, with users of similar use cases, like mMTC users in the same cell,
within the same group (slice). In the following, all the mMTC users belong to the same slice.}

The frame duration is $10$\:ms.  \textcolor{black}{The subcarrier spacing is $30$\:KHz, with $12$ subcarriers per block, making the block width $360$\:KHz.}
\textcolor{black}{The total number of PRBs is $275$~\cite{3GPP_5G_NR}}.

The simulations are conducted in MATLAB R2021a. We take the average of the metrics of interest over $10000$ runs.
\begin{table*}[t]
\caption{Per-block rates and the corresponding probabilities for users from the Republic of Ireland trace~\cite{trace_paper_5G}}
\label{Table-R_2}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
R (kbps) & 48 & 73.6 & 121.8 & 192.2 & 282 & 378 & 474.2 & 712 & 772.2 & 874.8 & 1063.8 & 1249.6 & 1448.4 & 1640.6 & 1778.4\\
\hline
$p_{1,k}$ & 0 & 0 & 0 & 0 & 0 & 0 & 0.01 & 0.05 & 0.11 & 0.13 & 0.14 & 0.18 & 0.06 & 0.11 & 0.21\\
\hline
$p_{2,k}$ & 0 & 0 & 0 & 0 & 0 & 0.01 & 0.02 & 0.06 & 0.13 & 0.14 & 0.2 & 0.21 & 0.07 & 0.09 & 0.07\\
\hline
$p_{3,k}$ & 0.01 & 0 & 0 & 0 & 0 & 0.01 & 0.01 & 0.02 & 0.06 & 0.13 & 0.17 & 0.18 & 0.08 & 0.18 & 0.15\\
\hline
$p_{4,k}$ & 0 & 0 & 0 & 0 & 0 & 0.02 & 0.03 & 0.13 & 0.06 & 0.2 & 0.32 & 0.11 & 0.01 & 0.09 & 0.03\\
\hline
$p_{5,k}$ & 0 & 0 & 0 & 0 &  0 & 0 & 0.04 & 0.07 & 0.13 & 0.17 & 0.22 & 0.2 & 0.05 & 0.06 & 0.06\\
\hline
$p_{6,k}$ & 0 & 0 & 0 & 0 & 0.01 & 0.03 & 0.11 & 0.12 & 0.19 & 0.15 & 0.15 & 0.12 & 0.05 & 0.04 & 0.03\\
\hline
\end{tabular}
\end{table*}
\begin{figure*}[t]
\begin{minipage}{0.33\linewidth}
\centering
\includegraphics[width=\textwidth]{pdf1.eps}
\caption{The probability density function of inter-arrival times for uniform inter-generation times.}
\label{fig:pdf1}
\end{minipage}
\begin{minipage}{0.33\linewidth}
\centering
\includegraphics[width=\textwidth]{pdf2.eps}
\caption{The probability density function of inter-arrival times for B. Pareto inter-generation times.}
\label{fig:pdf2}
\end{minipage}
\begin{minipage}{0.33\linewidth}
\centering
\includegraphics[width=\textwidth]{cv_lambda.eps}
\caption{Traffic variability at gNodeB vs. generation rate.}
\label{fig:cv_lambda}
\end{minipage}
\end{figure*}

\subsection{Impact of channel characteristics}
First, we consider the impact of channel conditions of mMTC users on the traffic pattern at gNodeB. We do this for homogeneous users, i.e., in a given scenario all mMTC users have i.i.d. traffic generation patterns. In the first case, users have uniform inter-generation times, whereas in the second their inter-generation times are drawn from a Bounded-Pareto distribution.

\subsubsection{Inter-arrival times for uniformly distributed inter-generation times}
In the first scenario, the inter-generation times are uniform, and the users of interest belong to type-1, type-3, and type-5 users (Table~\ref{Table-R_2}). The total number of users is $n=1500$. The number of generated packets at once is drawn uniformly from the interval $(10,20)$.
We consider the outcomes for three different inter-generation rates: $\lambda_1=0.01 s^{-1}$, $\lambda_2=0.05 s^{-1}$, and $\lambda_{3}=0.1 s^{-1}$. Table~\ref{Table-uniform} shows the results for the first, second, and third moments of the inter-arrival times at gNodeB in all the scenarios. Table~\ref{Table-uniform} shows also the coefficient of variation of the inter-arrival times at gNodeB, defined as
\begin{equation}
c_v(X)=\frac{\sqrt{\text{Var}(X)}}{\mathbb{E}[X]},
\end{equation}
which serves as the most suitable quantification for the variability of a stochastic process.
As can be observed from Table~\ref{Table-uniform}, the statistics for the same inter-generation rate are almost identical for all three user types. This implies that the \emph{channel statistics of the mMTC users have almost no impact on the traffic pattern at gNodeB}. The second thing to observe is that increasing $\lambda$ decreases the inter-arrival time of the traffic at gNodeB. This is expected as decreasing the time between generation instants will decrease the inter-arrival time as well.

\subsubsection{Inter-arrival times for Bounded Pareto inter-generation times}
\textcolor{black}{In the second set of results, the inter-generation times are Bounded-Pareto, characterized by the shape parameter $\alpha$, and minimum and maximum values $L$ and $H$, respectively. In all cases that follow, $\alpha=1.95$, whereas $L$ and $H$ are adjusted to achieve a given average inter-generation time. \textcolor{black}{When $\lambda=0.01 s^{-1}$, $L=48.75$ and $H=10000$. For $\lambda=0.05 s^{-1}$, $L=9.81$ and $H=2000$. Finally, for $\lambda=0.1 s^{-1}$, $L=4.905$ and $H=1000$.}}

\textcolor{black}{Table~\ref{Table-Pareto} shows the results for the same cases corresponding to those of Table~\ref{Table-uniform}, but for different user types (2, 4, and 6 now). Similar conclusions apply regarding the lack of impact of channel characteristics on the network traffic pattern. The variability is higher in this case (the value of the coefficient of variation can go up to $4.83$). The other thing to observe in this case as well is that \emph{increasing the traffic generation rates does not increase the coefficient of variation of inter-arrival times at gNodeB for any mMTC user type.}}

Fig.~\ref{fig:pdf1} illustrates the probability density function of inter-arrival times at gNodeB for user-type 1 and $\lambda=0.01 s^{-1},$ when inter-generation time is uniform, whereas Fig.~\ref{fig:pdf2} depicts the pdf for Bounded-Pareto inter-generation times for the same user type and the same average inter-generation time. Comparing Figs.~\ref{fig:pdf1} and~\ref{fig:pdf2}, we can observe that in the second case there are instances when the inter-arrival time can be $20\times$ higher than the maximum inter-arrival time in the first case. This is due to the high variance of Bounded Pareto distribution, classifying it as a  heavy-tailed distribution.
\noindent\emph{Note:} In Figs.~\ref{fig:pdf1} and~\ref{fig:pdf2}, for better visibility of the graphs, we have shown the results for $15$ mMTC users only.

The fact that the traffic pattern is \emph{almost completely insensitive} on the channel conditions of mMTC users has an important practical implication. Namely, network planning can be conducted based solely on the number of users and their traffic, but not on their channel characteristics. The reason lies in the nature of mMTC traffic, in which packets are small and traffic is relatively sparse, making inter-arrival times non-dependent on the per-block rate distribution.

\begin{table}[t]
\caption{The first three moments and the coefficient of variation for user types 1, 3, and 5, for three uniform inter-generation times}
\label{Table-uniform}
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
$\lambda_1=0.01 s^{-1}$ & $\mathbb{E}[X]$ & $\mathbb{E}[X^2]$ & $\mathbb{E}[X^3]$ & $c_V$ \\
\hline
user 1 & 0.068 & 0.037 & 2.95 & 2.59\\
\hline
user 3 & 0.068 & 0.037 & 2.95 & 2.59 \\
\hline
user 5 & 0.068 & 0.037 & 2.95 & 2.59 \\
\hline
\hline
$\lambda_3=0.05 s^{-1}$ & $\mathbb{E}[X]$ & $\mathbb{E}[X^2]$ & $\mathbb{E}[X^3]$ & $c_V$ \\
\hline
user 1 & 0.0136 & 0.0015 & 0.0236 & 2.59 \\
\hline
user 3 & 0.0136 & 0.0015 & 0.0236 & 2.59 \\
\hline
user 5 & 0.0136 & 0.0015 & 0.0235 & 2.59 \\
\hline
\hline
$\lambda_5=0.1 s^{-1}$ & $\mathbb{E}[X]$ & $\mathbb{E}[X^2]$ & $\mathbb{E}[X^3]$ & $c_V$ \\
\hline
user 1 & 0.0068 & $3.72\cdot 10^{-4}$ & 0.003 & 2.6 \\
\hline
user 3 & 0.0068 & $3.71\cdot10^{-4}$ & 0.003 & 2.59 \\
\hline
user 5 & 0.0068 & $3.71\cdot 10^{-4}$ & 0.003 & 2.59 \\
\hline
\end{tabular}
\end{table}
\begin{table}[t]
\caption{The first three moments and the coefficient of variation for user types 2, 4, and 6, for three Bounded Pareto inter-generation times}
\label{Table-Pareto}
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
$\lambda_1=0.01 s^{-1}$ & $\mathbb{E}[X]$ & $\mathbb{E}[X^2]$ & $\mathbb{E}[X^3]$ & $c_V$ \\
\hline
user 2 & 0.0697 & 0.156 & 253.45 & 4.83\\
\hline
user 4 & 0.0697 & 0.156 & 253.29 & 4.83 \\
\hline
user 6 & 0.0697 & 0.156 & 253.28 & 4.83 \\
\hline
\hline
$\lambda_3=0.05 s^{-1}$ & $\mathbb{E}[X]$ & $\mathbb{E}[X^2]$ & $\mathbb{E}[X^3]$ & $c_V$ \\
\hline
user 2 & 0.014 & 0.0063 & 2.02 & 4.83 \\
\hline
user 4 & 0.014 & 0.0062 & 2.04 & 4.82 \\
\hline
user 6 & 0.014 & 0.0062 & 2.04 & 4.82 \\
\hline
\hline
$\lambda_5=0.1 s^{-1}$ & $\mathbb{E}[X]$ & $\mathbb{E}[X^2]$ & $\mathbb{E}[X^3]$ & $c_V$ \\
\hline
user 2 & 0.007 & 0.0016 & 0.25 & 4.83 \\
\hline
user 4 & 0.007 & 0.0016 & 0.24 & 4.83 \\
\hline
user 6 & 0.007 & 0.0016 & 0.25 & 4.83 \\
\hline
\end{tabular}
\end{table}

\subsection{Impact of generation rates on gNodeB traffic variability}
Next, we look at how the inter-generation rates ($\lambda$) affect the variability of gNodeB traffic. We consider heterogeneous users now.
Table~\ref{traffic} depicts the traffic characteristics of the mMTC users. In total, there are 1500 mMTC users in the cell.
The number of packets generated at once by mMTC is uniform with the values $N_i$ from the corresponding intervals in Table~\ref{traffic}. We look at the coefficient of variation of inter-arrival times at gNodeB.

Fig.~\ref{fig:cv_lambda} illustrates the results. The values on the x-axis correspond to the sum of $\lambda$ over all six types for single users. The first value on the x-axis is $0.01+0.02+0.03+0.011+0.019+0.015=0.0905 s^{-1}$. In the second case, we multiply by $2$ the generation rates of every user, hence $2\times$ higher value of the x-axis. This is repeated $6\times$ (hence $6$ values in the plot of Fig.~\ref{fig:cv_lambda}). So, the highest value on the x-axis corresponds to traffic generation rates that are $6\times$ higher for all the users than those corresponding to the basic scenario from Table~\ref{traffic}.

The interesting outcome from Fig.~\ref{fig:cv_lambda} is that the variability of gNodeB traffic is oblivious to the traffic generation rates of mMTC users ($c_V$ remains almost unchanged at all with the increase in traffic generation rates).
\begin{table}[t]
\caption{Basic scenario with heterogeneous mMTC users}
\label{traffic}
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
Users & $\lambda_i$ & $N_i$ & inter-gen. time dist. & number of users\\
\hline
type 1 & 0.01 & (10,20) & uniform & 250\\
\hline
type 2 & 0.02 & (10,15) & deterministic & 250 \\
\hline
type 3 & 0.03 & (10,25) & exponential & 250 \\
\hline
type 4 & 0.011 & (10,20) & deterministic & 250\\
\hline
type 5 & 0.019  & (10,15) & exponential & 250 \\
\hline
type 6 & 0.015  & (10,15) & uniform & 250\\
\hline
\end{tabular}
\end{table}

\subsection{Impact of number of users on gNodeB traffic variability}
Having shown the (lack of) impact of traffic generation rates on the variability of network traffic, we look next at how increasing the number of users affects the coefficient of variation of inter-arrival times at gNodeB. We consider the same basic scenario from Table~\ref{traffic}, with one exception. The number of users changes from $50$ to $300$ for each user-types, i.e., the total number of users ranges from $300$ to $1800$. Fig.~\ref{fig:cv_n} shows the coefficient of variation for different number of users. As can be observed from Fig.~\ref{fig:cv_n}, increasing the number of users increases the coefficient of variation of inter-arrival times \textcolor{black}{almost linearly}.

\subsection{Impact of mMTC distance on traffic characteristics}
Having looked at the impact the generation rate and the number of mMTC users have on network traffic variability, we proceed with investigating the impact of distance of mMTC from the base station. The other data are from Table~\ref{traffic}. We consider four scenarios to that end. In Scenario~1, each mMTC user is placed uniformly in the range $(0.1,0.5)$\:km. In Scenario~2, the range where mMTC are is $(0.1,1)$\:km (again uniform), whereas in Scenario~3 every device is placed uniformly in the range $(0.1,2)$\:km. Finally, in Scenario~4, the position of each mMTC user is drawn uniformly from the range $(0.1,3)$\:km. Needless to say, but mMTC users' positions are fixed.

Fig.~\ref{fig:cv_d} depicts the values of the average inter-arrival times and their $c_V$ for the four scenarios. The impact of the distance is almost invisible on the gNodeB traffic characteristics. The reason is that the propagation delay is
very low and fixed for every device (mMTC position is fixed), making very little or no impact at all on the network traffic pattern.

\textcolor{black}{\subsection{Validation on a dataset}
In the last part of this section, we validate our theoretical result for the inter-arrival time at gNodeB. We use a trace of data from $21$ IoT devices as input parameters~\cite{IoT_trace}. More explanations on this dataset can be found in~\cite{IoT_trace_paper}. For every device, we randomly choose a user type and then the corresponding per-block rate from the other trace (Table~\ref{Table-R_2}). From the dataset~\cite{IoT_trace}, it can be observed that data transmission follows different patterns at different devices. In some of these, data are sent uniformly several times during $1$\:s, whereas in others every $5$\:s. There are no data available on the number of transmitted packets per time. We assume that 30 packets are transmitted at once. The other parameters are as in the previous scenarios. All the users are within the same cell. We observed that the coefficient of variation of inter-generation times for all the users is in the range $(0.5,0.6)$. Fig.~\ref{fig:cdf_trace} shows the CDF of inter-arrival times at gNodeB from these users. We model the behavior of every user in the system, to the best possible extent, by the corresponding distribution of inter-generation times (the number of packets is fixed whenever there is a transmission). Then, we use Eq.(\ref{eq:theorem_5}) to derive the CDF of the inter-arrival time at gNodeB. Fig.~\ref{fig:cdf_trace} shows this curve as well. As can be observed from Fig.~\ref{fig:cdf_trace}, our model can predict quite accurately the inter-arrival time distribution in this realistic scenario, despite the fact that the model may not perfectly capture all the intricacies of the generation process on the users' side.}
\begin{figure*}[t]
\begin{minipage}{0.33\linewidth}
\centering
\includegraphics[width=\textwidth]{cv_n.eps}
\caption{Traffic variability at gNodeB vs. number of users.}
\label{fig:cv_n}
\end{minipage}
\begin{minipage}{0.33\linewidth}
\centering
\includegraphics[width=\textwidth]{cv_distance.eps}
\caption{Traffic statistics at gNodeB for different distances.}
\label{fig:cv_d}
\end{minipage}
\begin{minipage}{0.33\linewidth}
\centering
\includegraphics[width=\textwidth]{cdf_trace.eps}
\caption{The CDF of the inter-arrival times from the trace and theory.}
\label{fig:cdf_trace}
\end{minipage}
\end{figure*}

\section{Related Work}
\label{sec:related}
While the traffic a single user generates, be it an enhanced Mobile Broadband (eMBB), URLLC, or mMTC, can be determined relatively easily depending on the application/service, predicting the traffic pattern (inter-arrival time and size distribution) at the base station from a (usually) large number of mMTC devices is not straightforward. To our best knowledge, there are no other works that can predict analytically the exact distribution of the traffic pattern at the base station.
Regarding the traffic distribution at single mMTC users, the general assumption is that inter-generation times for sparse traffic are either periodic (i.e., deterministic)~\cite{p14}, uniform~\cite{Hao19},~\cite{Alhussien20}, or exponential~\cite{Ortiz} for sparse traffic, and for mMTC users with more intense traffic, the number of packets underlie a beta distribution~\cite{Wang2020}, whereas the inter-generation time is governed by a Pareto distribution~\cite{Hao19}. In all these models, the sizes of the packets are fixed, and very small. However, in those works, there are no indications as to what distribution the traffic arriving from a large number of mMTC devices to the base station might have. On the other hand, in our paper, we assume a general distribution for the inter-generation times as well as for the number of packets generated at a single mMTC unit, and derive the distribution of the traffic pattern at gNodeB. Furthermore, for most of these single-mMTC traffic distributions,
we derive the traffic distribution at gNodeB. We consider as special cases the inter-generation times being deterministic, uniform, exponential, and Pareto distributed. While one of these works~\cite{Wang2020} assumes the number of generated packets underlies a beta distribution, our approach is more general, as the number of packets underlies any generic distribution, and it can be easily adjusted to capture this special case.

Predicting certain parameters of the traffic arriving at the base station using machine learning is done in~\cite{Weerasinghe19} and~\cite{Soraa20}. In~\cite{Weerasinghe19}, the authors propose a supervised learning approach that predicts the number of arrivals and when a burst of data will arrive for the number of packets that follow a beta distribution. However, the service type of interest in~\cite{Weerasinghe19} is URLLC. On the other hand, in our paper, we derive the exact distribution of both the number of packets arriving at the base station and the inter-arrival time between two data batches for general traffic generations at mMTC users. The main contribution of~\cite{Soraa20} is proposing a machine learning-based model for predicting possible occurrences of congestion under bursty traffic conditions, where the latter is modeled with a beta distribution. Contrary to~\cite{Soraa20}, our work is more general because deriving the exact distribution of the traffic pattern at gNodeB can also predict whether network congestion is expected to occur for any type of mMTC data distributions.

\section{Conclusion}
\label{sec:conclusion}
In this paper, we have derived the inter-arrival time distribution of the traffic from different mMTC users within the cell to the base station. We assumed a generic traffic pattern at each mMTC user, specified by a general inter-generation time distribution and a general number of packets transmitted at once.
\textcolor{black}{Conducting simulations on a real-life trace, results show that
the variability of the traffic pattern increases
when increasing the number of mMTC users, but
it is not affected by the increase in traffic generation rates.}
In our future work, we plan to characterize the traffic pattern of the other two service types in 5G - URLLC and eMBB.

\bibliographystyle{ieeetr}

redpajama_set_name: RedPajamaArXiv


"For dynamic web pages, do not ever use relative urls in your navigation.
Always, but always, use absolute urls only.
This entry was posted in Obiter dicta and tagged web design on April 24, 2011 by Carl D.
Hey, congrats on the growth of the empire. I see you have "Staff Writers" now. Minions are good.
All good websites have Staff Writers.


text:
Presiding Elder
Reverend Dr.
Alvin T. Durant
Alvin Tyrone Durant, a native of Richmond, Virginia, has always had the church as a foundation in his life. His faith in God began early while growing up as a member of the Hood Temple A. M. E. Zion Church in Richmond, Virginia where he participated diligently in its ministries and activities. His devotion to the church, coupled with the influence and guidance of his family and pastor, eventually led him into God's service.
He attended the public schools of Richmond where he graduated from John Marshall High School. In 1973, he entered Livingstone College in Salisbury, North Carolina where he earned a Bachelor of Science Degree. Soon after, he received his Master of Divinity from the School of Theology at Virginia Union University. Dr. Durant's accomplishments did not end there. He was among the first group of students to enter the Doctor of Ministry degree program at Howard Divinity School in Washington, DC, receiving his Doctorate in 1982.
For forty years Dr. Durant has been in the pastoral ministry of the African Methodist Episcopal Zion Church. He has served at the Lee's Chapel A.M.E. Zion Church, Chesterfield, Virginia, 1980-1981; David Temple's A.M.E Zion Church, Courtland, Virginia, 1981-1983; St. Mark A.M.E. Zion Church, East Chicago, Indiana, 1983-1993 and Mother Zion A.M.E. Zion Church, New York, New York, 1993-2005. His appointment as pastor of Mother Zion African Methodist Episcopal Zion Church, the "Mother Church of Zion Methodism," though filled with many new challenges and responsibilities, was a tremendous honor and privilege. In July 2005, Dr. Durant was appointed by Bishop Warren M. Brown the pastor of Union Wesley A.M.E. Zion Church. He passionately served Union Wesley AME Zion Church faithfully for fourteen years, and in May 2019 was appointed by Bishop W. Darin Moore, Presiding Elder of the Washington District African Methodist Episcopal Zion Church. There are fifteen churches on the Washington District.
Dr. Durant has served the African Methodist Episcopal Zion Church in several capacities. This includes the Connectional Budget Board of the AME Zion Church, Washington District Director of Conference Studies and Chairman of the Finance Committee, Philadelphia and Baltimore Conference Chairman of the Finance Committee and Instructor for Conference Studies, and Mid Atlantic Episcopal District Dean of the Leadership Training Institute. He currently serves as a member of the Finance Committee, Chairman of the Holy Orders Committee, Chairman of Conference Trustees Philadelphia and Baltimore Conference, and Mid Atlantic Episcopal District Chairman of Conference Trustees.
Dr. Durant has an untiring commitment and dedication to community service. He is a member of Phi Beta Sigma Fraternity, Incorporated, acclaimed as one of the Outstanding Men in America by the Atlanta Jaycees, and is a former professor at the Maple Spring Baptist Bible College and Seminary. On February 9, 2012, Dr. Durant was inducted into the Livingstone College Leaders Hall of Fame. This prestigious award is bestowed upon individuals who have given tirelessly of themselves as they serve their communities and/or their profession and have demonstrated the qualities of a servant-leader.
Dr. Durant has a strong belief and dedication to family. He is married to the former Brenda A. Pace of Tuskegee, Alabama, and they are the proud parents of one daughter, Nicole Dominique, one son Alvin Le'Mon, and daughter-in-law Shana. They are also the proud grandparents of one grandson, Ashton Josiah and two granddaughters, Dahlia Rose and Sydney Ann.


text:
Mike Buchanan brings a private prosecution against Joseph Spitzer, a doctor who's circumcised thousands of boys
June 25, 2017 By Mike Buchanan 26 Comments
Ending the non-therapeutic circumcision of male minors – Male Genital Mutilation, MGM – in the UK has been the leading campaign of J4MB over the past two years. We have considerably increased public awareness that MGM is illegal, being at least Actual Bodily Harm under the Offences Against the Person Act 1861, and probably Grievous Bodily Harm. It would require a parliamentary override for MGM to be legal, and that has never existed. No exemptions to the law in England and Wales are permissible on religious or cultural grounds.
Mike Buchanan will tomorrow present a 'written information' – a document outlining the details of a private prosecution, here – at Highbury Corner Magistrates' Court, charging Joseph Spitzer, a prominent London-based doctor and mohel (Jewish ritual circumciser), with inflicting Actual Bodily Harm, a criminal offence under the OAPA 1861. Spitzer claims to have circumcised thousands of boys over his 35-year 'career', presumably most of them at just eight days of age, in line with Jewish tradition.
Last December I confronted Dr Spitzer at his workplace – here (video, 3:00). A Voice for Men then added him to its Known Genital Mutilators directory – here.
In a press release to be emailed shortly to leading journalists, Mike Buchanan states:
Male Genital Mutilation is illegal in the UK, it's a human rights violation under UN and EU conventions, and it's equally as unethical as Female Genital Mutilation (Brian D Earp, University of Oxford ethicist). It results in physical and sometimes psychological harm, and it has no place in modern Britain.


Little Witch Academia is a Japanese anime franchise created by Yoh Yoshinari and produced by Trigger. The original short film, directed by Yoshinari and. Looking for information on the anime Little Witch Academia (TV) (Little Witch Academia)? Find out more with MyAnimeList, the world's most active online anime. Akko enrolls at the Luna Nova Witchcraft Academy. Ordinary girl Akko enrolls at Luna Nova Witchcraft Academy, yearning to become a witch just like her idol, Shiny Chariot. The enthusiastic but somewhat under-talented Akko, along with friends Lotte and Sucy, trains to be a witch at.
Akko's unconventional ideas don't go over well at the Magical Academy, but when a dragon is unleashed, she may be the only witch who can save the day. Netflix Adds Little Witch Academia Episodes (Aug 15, ) Netflix Debuts Little Witch Academia TV Anime's 1st 13 Episodes on June 30 (May 23, ). Animation Akko Kagari is an ordinary girl who joins the renowned witch academy for witch girls, Luna Nova Academy. Photos. Little Witch Academia ( ).
Uncover the mysteries at Luna Nova Academy and experience the magical world of Studio Trigger style Japanese animation in Little Witch Academia: Chamber. 15 hours ago Little Witch Academia: Chamber of Time is one-third of a good game, held back by convoluted systems and tiresome grind. Read our full. Little Witch Academia is an anime franchise created by Yoh Yoshinari and through Kickstarter, titled Little Witch Academia: The Enchanted Parade, was. 15 May Uncover the mysteries at Luna Nova Academy and experience the magical world of Studio Trigger style Japanese animation in Little Witch. Little Witch Academia (リトルウィッチアカデミア Ritoru Witchi Akademia) is an anime adaptation to the original eponymous film and its sequel, The Enchanted.
Atsuko Kagari (アツコ・カガリ Kagari Atsuko), more commonly known by her nickname Akko (アッコ), is the main protagonist of Little Witch Academia. She is a first. 15 May Metacritic Game Reviews, Little Witch Academia: Chamber of Time for PlayStation 4, Uncover the mysteries at Luna Nova Academy and. Description. Uncover the mysteries at Luna Nova Academy and experience the magical world of Studio Trigger style Japanese animation. Join Akko and her. A description of tropes appearing in Little Witch Academia. Witch-in-training Atsuko "Akko" Kagari dreams of following in the footsteps of Shiny Chariot, .
15 May Bandai Namco's Little Witch Academia: Chamber of Time - an adaptation of the popular anime series - is an action-RPG which allows fans the. Novice witch discovers her inner powers in anime series. Read Common Sense Media's Little Witch Academia review, age rating, and parents guide. This entry is WIP. Any requests for help are welcome] About Little Witch Academia (also known as リトルウィッチアカデミア Ritoru Witchi Akademias) a half- hour. Set after episode seven of season three of Rick and Morty and episode 25 of Little Witch Academia. A mysterious traveler, a Morty, shows up outside Luna Nova.


Distance from Midway International Airport to Will Rogers World Airport is 691 miles / 1112 kilometers / 601 nautical miles.
Estimated flight time from Midway International Airport (MDW) to Will Rogers World Airport (OKC) is 1 hour 48 minutes.
There is no time difference between Chicago and Oklahoma City.
Flight map from Chicago to Oklahoma City.
Chicago (MDW) → Oklahoma City (OKC).


text:
namespace XCommon.CodeGenerator.TypeScript
{
	public interface ITypeScriptEntityWriter
    {
		void Run();
    }
}

redpajama_set_name: RedPajamaGithub


text:
Hong Kong axed from economic freedom index after years at top spot, now ranked 107th as part of China
by Candice Chau 13:31, 4 March 2021 16:56, 4 March 2021
Hong Kong is now counted as part of China and has been excluded from a US think tank's economic freedom index. The Heritage Foundation scrapped the city's 2021 listing saying that its economic policies are "ultimately controlled from Beijing."
In the 27th edition of the Index of Economic Freedom, Hong Kong is no longer listed separately despite topping the chart for 25 consecutive years.
File photo: AFP/Srtringer.
Hong Kong has been ranked the freest economy in the world since 1995, until it was overtaken by Singapore in 2020 after the foundation cited "intensifying uncertainties related to security issues" due to the government's response to the 2019 anti-extradition bill protests.
The Hong Kong government often boasted of the accolade in promotional material for the city. In 2019 – after Hong Kong became the only economy that had scored over 90 – Financial Secretary Paul Chan said: "This achievement reaffirms the Government's steadfast commitment in upholding the free market principles over the years."
In this year's index, the conservative think tank explained that the ranking would only include "independent countries where government exercise sovereign control of economic policies."
"No doubt both Hong Kong and Macau, as Special Administrative Regions, enjoy economic policies that in many respects offer their citizens more economic freedom than is available to the average citizen of China, but developments in recent years have demonstrated unambiguously that those policies are ultimately controlled from Beijing," the report read.
The Index's website had also removed Hong Kong's standalone page, and no results were shown when "Hong Kong" was typed in the search bar.
Index of Economic Freedom 2021. Photo: Heritage Foundation, via website screenshot.
Singapore topped the ranking again this year, with a score of 89.7.
China scored 58.4, and was ranked 20th among the 40 economies in the Asia-Pacific region, and 107th in the world among the 178 countries examined.
The report also stated that China's score was dampened by a decline in trust and destabilised global relationships due to issues such as "restrictions on Hong Kong's autonomy, the persecution of China's Uighur minority, and Beijing's cyberwarfare activities."
'Political bias'
Speaking in Beijing, Hong Kong finance chief Paul Chan said he did not agree with the foundation's analysis: "When they arrived at that decision, they must have been clouded by their ideological inclination and political bias," he said according to RTHK.
Billy Mak, associate professor of the Department of Finance and Decision Sciences at the Hong Kong Baptist University, told HKFP that the think tank's decision to stop listing Hong Kong as an independent economy could harm the study's credibility: "This is an academic issue, if you use the previous classifications, you can conduct different comparisons between countries and regions over the years, but now that you suddenly changed the evaluation criteria, there may be discrepancies between the new and old data."
Mak added that by not listing Hong Kong as a separate economy in the index, it may be difficult for researchers or the government to understand and compare the situation in Hong Kong to those of other regions.
HKFP has reached out to the government for comment.
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Tagged: Billy Mak, Heritage Foundation, Index of Economic Freedom, Macau, Singapore
Candice Chau
Candice is a reporter at Hong Kong Free Press. She previously worked as a researcher at a local think tank. She has a BSocSc in Politics and International Relations from the University of Manchester and a MSc in International Political Economy from London School of Economics.
More by Candice Chau
'Right now, truth is subversive in Hong Kong': Local media fearful after arrests, newsroom raids
China says UK spy claims against Christine Lee a result of watching 'too many 007 movies'