Medicine is becoming increasingly corporatized, and some doctors are very excited when they get offered a job at these hospitals.
They are promised a fat salary , and the job seems to be all they could ask for ! The hospital takes care of all the administrative hassles, so that the doctor can focus on taking care of his patients – and the hospital promises them that they'll have a long line of patients clamouring to see them. Doctors are not good managers, and are very happy if someone else does all the work of attracting patients; billing them; and providing the secretarial support they need, so they can do what they enjoy – helping their patients get better. Since this is all done under one roof, using high quality new equipment, the doctor does not need to run around from clinic to clinic to get more patients – and does not need to offer kickbacks to GPs to refer patients to him either ! This seems to be a win-win situation, and it often is for the honeymoon phase, which lasts a few years. At this time, the hospital is new, and they need senior brand name doctors to attract patients. This is why they offer great signing packages when they launch their hospital with a bang, and everyone is delighted in the beginning.
However, in a few years, once the hospital starts becoming profitable, they start showing their true colours. These are for-profit hospitals, which have a clear focus – how do we maximize our revenues ? They understand only one thing – numbers, and soon the management starts singing a new tune. They talk about targets which the doctors need to meet – bed occupancy; number of patients admitted; revenue earned; number of CT scans ordered ; and number of operations done. The focus has now shifted from providing high quality care to patients to earning more money.
Doctors who are high performers and who run the hospital's profit centers are treated as blue eyed boys. However , the rest are told to shape up or to get out. Most doctors buckle under this pressure, and will fall into line quickly, in order to save their jobs. They start doing unnecessary surgery and ordering too many tests, to get their numbers upto speed, so the management can give them a pat on their back and a bonus as well.
" requested " to improve their performance. If they don't improve, their contract is not renewed. Many are insulted , until they are forced to leave ( for example, the consulting time slots allotted to them are curtailed to one slot per week, so they have very few patients) . It's easy for the management to make a doctor's life miserable ! As a friend of mine who is a senior surgeon said - " I feel I have been treated as a condom – used and then thrown !"
Hospitals treat all doctors as being interchangeable – they are viewed as being cogs in a wheel. If a senior doctor becomes too expensive for them, they get a junior to replace him, because the "cost to company" is much less - and they really don't care about clinical acumen, surgical dexterity or bedside manner, because these are not numbers they can measure !
The fact that hospitals are focused on maximizing their profit does not mean that they are villains. This is to be expected – after all, they are answerable to their share-holders and cannot afford to fritter away money on expensive doctors ! The real problem is that doctors are so simple-minded. They seem to know only one thing – how to take care of patients. While they do a very good job at this, most of them have very few real-world skills, so it's very easy for the management to take them for a ride ! They have no unity and it's very easy to play one against the other.
that he is between a rock and a hard place ! By the time he is 50 , he cannot afford to retire and neither can he afford to start afresh. He is forced to swallow his pride and accept all the rubbish which is doled out to him. He is furious about the fact that he spent the best years of his life working for the hospital and helping them to become successful, and is now being treated like dirt. He retaliates, by taking out his frustrating and angst on his staff – and his patients ! This is the major reason why morale in corporate hospitals is so poor. It's only their façade which is beautiful , but because doctors feel devalued and unappreciated, they do not put their heart and soul in taking care of their patients.
Traditionally surrogacy has been reserved as a treatment option for women whose uterus was absent or was irreparably damaged . It is an expensive treatment option , which involves a lot of coordination as well as legal issues to make sure that the child is handed over to the intended parents. However, we are now seeing an interesting trend , so that there are women who are no longer using surrogacy for medical reasons , but for social reasons !
These are women who know that they are able to carry the pregnancy themselves, but who decide for social reasons not to get pregnant themselves , but to use a surrogate instead. They have a wide variety of reasons for choosing this route. Some are very busy and successful career women, who don't want to take a break in their professional career , as this will harm their chances of becoming the CEO. Others are models who don't want their figure to altered as a result of pregnancy. Some feel it's just too stressful for them to go through pregnancy , and if they don't need to do it themselves , and can get someone else to carry a baby for them, then why shouldn't they use this option ?
This raises a number of interesting ethical issues ! Is it right for the doctor to accept such requests, even when there is no valid medical reason for doing surrogacy ? Or is the doctor being very presumptuous by refusing such requests ? After all , these are intelligent, educated successful women, who know their own mind . If this is what they have decided they want the doctor to do for them, is it ethical on the part of the doctor to reject these requests, just because he feels they are not medically appropriate ?
After all, if a decision is made by a patient , doesn't the doctor need to respect her autonomy ? If there no downsides to doing surrogacy for social reasons , shouldn't patients be allowed to choose for themselves ? These are thorny issues , and we don't have all the answers as yet .
It's interesting to see how reproductive technology which was originally developed for treating medical problems is now increasingly being used for social reasons . For example, egg banking , which was originally invented as a method to preserve the fertility potential of young women with cancer, is now far more commonly used by women who want to postpone childbearing for nonmedical reasons !
When patients do an IVF cycle, the only thing really care about is the outcome – they just want a baby ! If life were perfect , then every time the patient had a perfect IVF cycle , she would get pregnant . Unfortunately, human reproduction is not an efficient enterprise , and the outcome of an IVF cycle is always uncertain. This is because biological processes cannot always be influenced by the doctor and we have very little control over what will happen to the embryo after we transfer it into the uterus. The sad reality is that we cannot manipulate the process of implantation This is not something which must patients want to hear . It's an uncomfortable feeling knowing that no matter what you do, you can never be sure what the result of the IVF cycle will be – each IVF cycle is effectively a gamble , which you may have to repeat multiple times , until you achieve success.
This is why it's important to differentiate between processes and outcomes . While outcomes cannot be controlled , the process itself can be, and a good IVF clinic will follow good processes. Now if life were always, fair good process would always lead to good outcomes, so that everyone's happy. Unfortunately, good processes can lead to bad outcomes , and an IVF cycle can fail because of circumstances which are beyond the doctor's control.
While mature patients understand this , most patients are not sophisticated enough to appreciate this, and if the IVF cycle fails , they are understandably upset and frustrated . They will often vent their anger on the doctor , complain about his clinic, and even write bad online reviews that he's incompetent and has not done a good job , even though the process he followed may have been perfect . Conversely, we do have bad IVF clinics , where patients do get pregnant, in spite of the fact that they follow bad processes . This is just plain dumb luck - and is particularly common in very busy clinics who will do lots of IVF cycles for young patients who don't even need IVF in the first place ! These clinics will obviously have a high success rate , as a result of which their reputation in the community will spread , because patients are not sophisticated enough to understand that the good outcome was not a result of good processes, but because of smart patient selection ! As a result, some of these doctors continue to fool some of the patients some of the time . Finally you have clinics which follow bad processes and therefore have bad outcomes. These will shut down in a few years – but only after having caused a lot of harm !
When a cycle fails, most patients cannot differentiate whether the bad outcome was because the doctor followed bad processes , or whether it was a result of plain bad luck, even though the processes were followed perfectly. Unfortunately many IVF doctors are not transparent ; and few patients are willing to invest the time and energy in order to understand the process which were used. Consequently, many clinics get away with fooling their patients , by preying on their ignorance .
Just like it's important for a good doctor to follow good processes it's equally important for a patient to become a good patient and do their homework , so they understand the difference between good processed and bad processes, no matter what the outcome !
Surrogacy means many things to many people. For infertile couples it's a chance to have a child. For working women, it's a shot at parenthood without a mid-career break. For surrogate mothers, it's big money. And, of course, for the lawyers and doctors, it's big business. Rashmi Pratap takes a look at the flourishing business of surrogacy.
A woman's age is always a sensitive topic - and this is true, no matter how old she is. If she's infertile, she is likely to become even touchier about this, because she's aware of the fact that she has a biological clock which she cannot rewind. As she ages, she no longer has all the time in the world in order to have a baby.
Is this the ultimate in outsourcing ?
Back-of-the-envelope calculations show that about 1 out of every 5 doctors passing out each year could be those who bought seats in medical colleges.
The huge amount of money which politicians can make out of capitation fees and management quotas has led to an explosion of medical diploma mills ( who are allowed to call themselves medical colleges) which churn out doctors who are poorly trained and poorly qualified. Sadly, the Medical Council of India has failed sadly in ensuring medical education standards are met - it's easy to make doctors buckle under political pressure.
All the more reason that patients need to do their homework before seeing a doctor. Just the fact that your doctor has a MD degree is no longer enough evidence of his competence !
Information Therapy can help you to save your life !
This article in the Times of India describes changing trends in babymaking.
I'd love to hear your comments !
Is it understandable that because we have high expectations from our children, if people want to have children with certain characters, we should allow them to use technology to do so ? After all, isn't this why men look for fair brides, so they can have fair children ? Isn't this technology a logical extension of fulfilling their desires ?
Or is this is a misuse of technology ? Are parents being greedy and too demanding ? Do "designer babies" represent a downward slippery slope where children are treated as a commodity which can me made to order ?
Can you postpone motherhood? Clinically, it is possible, Dr Aniruddha Malpani tells Amy Fernandes.
Red the article in Today's DNA newspaper !
I just completed a consultation with a single 38-year-old banker who wanted to freeze her eggs. She said, " Doctor , I have still not been able to find the right guy to marry , and I am not getting any younger . If at some point I do want to have babies, I want to freeze my eggs now so that I don't have to worry about the increased risk of birth defects or reduced fertility as I get older ".
I explained to her that we could freeze her eggs, but my concern was that she may have already missed the boat ! The right time to freeze eggs is when they are of good quality, and there are lots of them. From a biological perspective, this should be done when a woman is young - preferably at the age of 25 !
I understand it's unrealistic to expect 25 year old women to freeze their eggs . When you are young, and you have your whole life ahead of you, the probability of being able to find the right person , and then settling down and have a baby in the bedroom is extremely high ! However, if you postpone this decision until you cross 35 , even though egg freezing still remains a viable option , the reality is that the success rate of having a baby with these frozen eggs does drops dramatically if the freezing is done after the age of 38.
This is because ovarian reserve declines with age, and while the decline varies from patient to patient, it does mean that the success rate with egg freezing goes down . So what advice should a specialist give to a woman who wants to consider freezing her eggs ?
In the best of all possible worlds , the best time to freeze eggs is when you don't need to - when you're young and have lots of them ! However, in real life , there are very few women who will consider an option which is expensive - and one which they may end up not using at all.
The trouble is that when women are good candidates for egg freezing - between the ages of 30 to 35 , they have so much going on in their life , and rightfully feel that their chances of finding the right person and getting married are so high , that they still not willing to consider the possibility that they may have difficulty having a baby in the bedroom when they want one. The biological clock is still quite muffled at this stage in a woman's life ! This is why it's usually when they start crossing 37 that they start exploring all these additional options . They may have heard stories from their friends, who needed to do IVF to have a baby. This is wake-up call for them, and they are prodded into exploring their reproductive desires much more closely. While we are happy to do egg freezing for any woman, no matter what her age ( after all, he cannot become younger, and it's better to do it late than never !) , the reality is that they may not have such a great success rate , and it would have been much better for them to have done it earlier, rather than to have postponed making this decision for so long.
So should I freeze her eggs or not ? This is always a hard decision to make - especially for something which is elective treatment , and which can be quite expensive ! This is a decision which the patient has to make, and I am the kind of doctor who is very reluctant to tell patients what to do, I feel my job is to be a counselor , and allow them to come to their own decisions.
I help them to consider the pros and cons of both paths, and they suggest they take the path of least regret, so that they have no regrets afterwards.
This was my explanation. Suppose we do freeze your eggs , and then you find Mr Right the next month and then have a baby in your own bedroom - how much will you resent the fact that you spent your money on freezing your eggs and never used these frozen eggs ? On the other hand, suppose we don't freeze your eggs , and then you decide you want have a baby at the age of 42 , and learn at that time that you can't because your eggs have become too old. How much will you regret not having frozen your eggs when you were 38 ? These are very personal answers - and they will vary from person to person ! The best thing to do is to choose whichever path causes you the least regret ! These are obviously very individual decisions , which are not to be taken lightly , and it involves using both your head as well as your heart .
It's very easy for a doctor to say - This woman has come to me for egg freezing , so I am going to go ahead and do it for her - and from a doctor's perspective , it makes a lot of sense to do so, because doctors get paid for performing procedures ! From a financial point of view , it makes no sense for a doctor to say no to a patient who wants egg freezing. However, patients need to have realistic expectations of what we can do with these frozen eggs, which is why doctors need to counsel them and arm with them all the facts.
The woman came back after 3 days and said - " I know I am 38 and my eggs are old, but I am not getting any younger , and if I freeze my eggs , even if I do find someone and have a baby in my bedroom, these eggs will always be available to me as a backup option . In any case, even if I end up not using my frozen eggs. I won't feel bad about it. " I was happy to hear this, because it's crystal clear that she's thought through things logically and analytically , and then come to a decision which she is comfortable with.
Patients need to remember that egg freezing offers hope - not a guarantee. If patients have unrealistic expectations of medical treatment , they are likely to be unhappy with the doctor , and unhappy patients make for unhappy doctors.
The home pregnancy test showed two lines !
" Yes, it is a positive. I saw a positive hpt almost 4 years back, in 2009 January. If that embie had managed to become my baby, my child would have been three and a half years by now. I need to remember that little embie which gave me the strength to hold on to hope and keep trying. I am feeling so calm and serene. No overt emotions are there. After expecting this for a long time, I feel surprisingly numb. I am grateful that I had a positive. I know from now on it is a long journey and I wish I get the strength to face whatever comes my way. Surprisingly, I am not afraid of the future.
I take this opportunity to thank my Dr. I am a very difficult patient to deal with - both scientifically and emotionally. He was always kind and compassionate. He never made me feel that I am stupid or irrational. His team is as wonderful as he is. "
We are very excited and happy for Manju - and our prayers and best wishes are with her.
A new journey now begins !
Doctors think of themselves as being indispensable . Most have pretty high self-esteem – after all, in order to successfully get through medical college , you need a lot of physical stamina and mental energy. You need to perform well academically , in order to graduate , and if you are used to being in the top 10 % of our class, you start thinking of yourself as being special !
The fact that their patients are so dependent on them – and that they are willing to wait patiently for hours for their advice , it's easy to understand why doctors get used to being put on pedestal. It's hard to think of yourself as being only human, when you occupy such a superior position, which is why lots of doctors think that they play a role which is indispensable . They take pride in their academic knowledgebase as well as their professional skills , and feel that no one else can do quite as good a job as they do – all doctors have a superiority complex !
While it's true that doctors are valuable , it's also equally true that a lot of the stuff which doctors do can be easily and quickly done someone else who's a much less expensive resource ! Often, this person would do a better job , because he does it again and again, and becomes expert at doing it !
Doctors should try to make themselves redundant , so that they are able to delegate their duties – either to an assistant , or with the help of technology. This would allow them to free up their time , which they could then devote on things which they find more intellectually challenging. The use of physician extenders and physician assistants are a real world example of cleverly using human help in order to improve their efficiency .
Any doctor who develops tools and techniques to free up his time can then leverage this in order to improve his personal productivity. Since all doctors have similar pain points, this can also provide the nucleus of a valuable product or service , which lots of healthcare systems would be happy to pay for, in order to make their physicians more efficient !
It breaks my heart when I see what a poor job the Indian government does with educating patients. Sadly, even hospitals are no better - and there is precious little educational materials available in regional languages for Indian patients. This is a crying shame - for a country which prides itself on offering world-class medical facilities - and which is the IT powerhouse of the world.
Fortunately, when I see the initiatives taken by individuals to address this deplorable state of affairs, my faith in humanity is restored.
The private life of celebrities has always been an area of great interest to the average person , which is why they hog so much of the limelight. Their life is tracked with great interest, and anything which happens to them gets splashed all over the media. This is why whenever a celebrity does IVF , this is often front-page news . Readers and TV channel viewers want to know all the gory details. Why did they need IVF ? did they use donor eggs ? or a surrogate ?
Reporters have to do a lot of digging to get at the truth – and this has consequences - some of which are good , and some of which are not. The details of the celebrity's personal medical problems become the staple of cocktail party gossip – and there's lots of exaggeration and distortion. I'm sure most celebrities would rather keep this part of their life private. However, given the fact that they have chosen to be a celeb, this is one of the trade-offs they accept - the sacrifice of their privacy . What happens in their lives seems to be pretty much grist for all mills – and it all gets rehashed many times.
One of the advantages is that there is a lot more public awareness about IVF and the medical illnesses which it can be used to treat successfully. The acceptability of a lot of these procedures in society ( which originally were considered to be taboo topics ) increases considerably . Thanks to the media coverage and publicity which these stories generate, a lot of people believe that if a celeb can do surrogacy, so can I ! They start exploring this option , as result of which there are many more patients who start asking for these procedures. While increased awareness is always good , because patients are much more aware about what can be done to help them solve their problems , the downside is that that there are lots of inappropriate requests for these procedures , so that even patients who don't need such expensive and complex treatment start clamouring for them.
This is partly a result of the poor quality of reporting in the press . because it's not possible to provide enough medical background in these stories. Also, many reporters feel that because readers are not interested in the medical facts, they often provide only skimpy medical background.
However, thanks to these celebrities, a lot of these, topics have now become mainstream , so that it makes it much easier for IVF doctors to discuss some of these treatment options with patients because there's been so much coverage of them ? And, another side effect is that a lot of IVF doctors who treat celebrities become celebrities in their own right !
It's a fact of life that kickbacks, cuts and commissions have become an integral part of the healthcare system in India. In fact, this is one of the reasons why healthcare has become so sick. However, you disguise them - in the form of referral fees ( the euphemism which corporate hospitals use when they pay family physicians to send patients their way); or consultancy fees ( the dignified term which pharmaceutical companies use to incentivize doctors to prescribe their brands), commissions are ubiquitous have become an accepted norm in the medical profession.
A lot of this was started by pharmaceutical companies, who provide doctors with freebies in order to influence them to prescribe their drugs. Doctors have become so used to getting so much stuff for free from pharmaceutical companies - ranging all the way from trips to go overseas to cars and even apartments, that most of them take these gifts as their right. They feel they are entitled to them, and do not even question the morality of their behavior.
While all doctors know in their heart of hearts that it's unethical to take these kickbacks it's extremely easy for them to justify what they do !
All doctors, including the leading lights of the medical profession do it, so why shouldn't I ?
I can't afford to take a holier than thou attitude - how will I be able to afford to pay the capitation fee to send my children to medical college ?
Since everyone else in society is doing it, ranging from politicians to bureaucrats to policeman, so why should I be any different. Should doctors be held to a higher ethical standard?
While it's great to get free stuff, most doctors feel uncomfortable about taking kickbacks. Most of us do have a conscience, and while we learn to justify our behavior, taking kickbacks is not something which most doctors would rather do - they'd rather earn money honestly !
A lot of us fail to realize the harm these ubiquitous kickbacks do. For one thing they make the cost of medical care much more opaque , as a result of which patients often don't understand why certain tests are being advised; or why a particular brand of drug has been prescribed, even though it costs much more.
Patients are smart enough to know about the kickback system, because it's so pervasive. This causes them to start distrusting their personal doctor. They start treating the entire medical profession with suspicion, and this poisons the doctor-patient relationship. Patients no longer look upon doctors with high regards, and if they cannot trust their doctor, it becomes much harder for them to heal.
It's unfair on honest doctors as well. Everyone gets tarred with the same brush, so that patients automatically assumes that all doctors must be on the take !
I don't think there's any point in preaching to doctors that they should not be taking cuts and kickbacks because this is unethical. Good doctors know this - and bad doctors are going to improve.
We need to understand that cuts are given purely for financial reasons, and this is what we need to address. Cuts and kickbacks exists because there is a middleman - and the only way to cut out cuts would be to cut out the middleman. Thus, the reason a diagnostic center provides a kickback to doctors is to incentivize the doctor to refer patients to them. However, if diagnostic centers could reach out to patients directly, there would be no need for them to go through the middleman, and they would not need to provide these kickbacks anymore. This would help them to cut down the costs of their tests, thus helping patients as well.
A number of studies have shown that the healthcare industry is amongst the most corrupt in the world. This is a crying shame. The only way to fix the system is by encouraging openness and transparency and the internet provides us with a great platform in order to do this.
By making intelligent use of the internet, the healthcare insurance industry has a great opportunity to clean up the system. By publishing the costs of tests and medicines online, they can share this information with patients. Armed with this information, the patient has a much better sense of exactly how much each test should cost. He can then intelligently question the padded bills he gets, and by behaving as an intelligent consumer, he can get the most cost-effective testing done for himself, without having to depend on a referral by his doctor.
Agreed that it takes time for the system to change, but the good news is that it's possible for this to happen. And this is a great chance for us to start to clean up the system - something which is in everyone's best interests - patients, doctors, and society.
The Times of India recently reported that the Maharashtra Medical Council is taking action against doctors for advertising their skills.
Whether or not to allow doctor to advertise has been a thorny issue for many years – not only in India but in the US as well . The traditional approach has been that doctors are professionals , and that advertising is below their dignity because they're not businessman. The belief used to be that there was no need for good doctors to advertise, as patients would come to them because of their reputation.
However, we need to acknowledge that times have changed , and that we need to change with them. The purpose of advertising is not merely to flaunt the doctor's skills or to solicit patients - it's a very efficient mechanism of allowing patients to know what special skills doctors can offer ; which specialist are available; where they are available ; and how much they charge for their services. This is important information , which allows patients to select the right doctor for themselves. This is especially true in a democracy which is a capitalistic society , because patients have the freedom to choose . Patients are no longer passive recipients of medical care - they have become active consumers , who have the right to know what choices are available to them. By depriving them of this right , the Medical Council is actually doing them a disservice.
This does not mean that doctors should be given a free hand and be allowed to advertise at will. Some of the advertisements actually make me cringe , when I see how inaccurate and unreliable they are. Not only do they misguide patients, they also demean the dignity of the medical profession. However, that's no reason to throw the baby out with the bathwater . It's possible to find a middle path which creates a win-win situation for doctors and for patients.
The Medical Council can treat this as an opportunity, so that they can project themselves not just as a body which is responsible for governing the medical profession , but one which is enlightened enough to put patients first ! One way of doing this is by ensuring that the information which patients want is easily available to them - and the best of doing this is going online. Thus the Medical Council website should become the first website for patients to visit, whenever they have any questions about any medical problems.
If the Medical Council site provided Information Therapy , patients would then be able to make much better informed decisions. Not only could they learn about more about their medical problem , the Medical Council would also provide a list of all the registered bonafide doctors , who are qualified to provide care about their illness.
Doctors would be allowed to provide information about their services on the website – and because it's an official site, the information can be regulated and controlled . There would be standard guidelines which everyone would need to follow , to make sure there was a level playing field . This would be a much more ethical and transparent way of making sure that everyone's interests are adequately met.
Not only would patients be happy, because they would have all the information they require in one place, but good doctors would be happy as well , because they could refer their patients to this website .
By not allowing doctors to advertise at all , the Medical Council is doing a disservice , not only to patients, but to doctors as well. By preventing patients from getting access to information about specialized services , patients are left in the dark ,and are forced to rely on a family physician to refer them to the right doctor. Sadly, as we all know, doctors will often not refer patients to the best specialist, for extraneous reasons.
The Council is also doing a disservice to the thousands of junior doctors who are starting practice . Banning advertising is very good for senior doctors like me , who have a well-established practice, and lots of patients. It's easy for me to get more patients, thanks to word of mouth. But what about the young doctor who is just starting practice ? How is he going to be able to get patients unless he has access to a platform which will let patients know about his special skills ? Preventing advertising actually encourages kickbacks and commissions , because it prevent doctors from reaching out to patients directly. Doctors are then forced to incentivize middlemen to refer patients to them – and this will make the practice of giving cuts even more entrenched.
While it's true that advertising can be misleading, and can allow bad doctors to prey on helpless patients., if advertising is done properly, it can actually help to cut down healthcare costs . Doctors will be forced to be open and transparent - and in a competitive marketplace , patients will be able to know exactly what the fair price for a particular service is, so they know exactly how much they should be paying .
This is a great opportunity for the Medical Council to reinvent itself , and to keep up with the times , so that they can be of service not only to the medical profession , but to patients and society at large . The Medical Council can prove that the medical profession puts patients first , by providing a platform which all doctors and patients can use as a trusted source of information .
By taking this leadership role, the Medical Council can help to position India as a medical powerhouse of the world !
IVF treatment is complex and expensive. It needs a lot of coordination and monitoring.You have a busy life style and need to know in advance what is going to happen when, so you can plan your work around your IVF schedule. You cannot afford to forget your meds or miss your appointments !
MyFertilityDiary is a free app for android smartphones which can help you organise your life and manage your IVF treatment efficiently !
The unique heart of the app is the Treatment Calendar. This has prepopulated schedules for the commonest infertility treatments , including the various IVF and IUI protocols. You can modify this schedule this for your own treatment, as needed. The app is designed around a Calendar/Organiser which allows you to organize your life. You can set reminders, so you remember their appointments and their medications.
A lot of patients ask me for a second opinion after their IVF cycle fails. In order to try to help them, I need some basic medical data about their IVF cycles, such as: how many follicles did they grow; how many eggs were retrieved; and photos of their embryos. Some patients get very agitated and upset when I ask these questions because they often don't have this basic information. When I ask them why their documentation is so patchy and why they are so poorly informed, the standard answer is - This is all my doctor provided me. When I tell them that every good IVF clinic routinely provides a written treatment summary and embryo photos, they get defensive and say - I trust my doctor, so why do I need to know all this detail ?
There is a lot of criticism that surgeons do too many unnecessary operations and that all this overtreatment increases the cost of healthcare . Even worse, all this surgery doesn't help to improve patient outcomes – and can actually result in harm.
While there is truth in this criticism, we also need to understand that society itself is partly responsible for the sad state of affairs. After all, if you teach and train surgeons how to do surgery , and then you pay them to perform surgery, it's very irrational to expect them not to do surgery ! After all surgeons are homo economicus as well and will maximize their personal benefits .
It's not that they are greedy or unethical. They have a particular worldview , and often suffer from tunnel vision . All they are taught is how to operate and when you have a hammer in your hands, you see nails all over the places ! Unfortunately, very few surgeons are taught when not to operate !
It's possible to wax eloquent about the importance of medical ethics and why and how doctors need to be professionals , who should not operate when not required. However, we can talk till we are blue in the face , but we know that the reality will not change. Just like financial advisors will push a particular product , because they are paid to do so , doctors also are likely to respond in exactly the same manner! It's unfair to hold them to a superhuman standard , because this is extremely unrealistic and smacks of wishful thinking.
However, there is a solution – we need to develop clever ways in which we could incentivize surgeons not to do surgery ! There was a time in the US not too long ago, when repeat cesarean sections were the norm because obstetricians would get paid much more doing a cesarean section , rather than allowing vaginal birth ( VBACs). It's only when insurance company started to refuse to pay for repeat cesarean sections that the VBAC rate decreased from virtually zero percent in private hospitals to up to 50 percent - which is what the optimal number should be !
We need an entrepreneur who can come up with a clever solution which will help to ensure that surgeons do not benefit by doing unnecessary surgery. One solution is to slap penalties for doing unnecessary surgery , but this is something which is very hard to implement in real life . Clinical decisions are rarely made by following a cookbook protocol slavishly - we need to come up with more ingenious methods .
One of the best methods would be empowering patients with information , so they can discuss non-surgical treatment options with their surgeon ! We need to align incentives for both patients and surgeons – and find ways to pay surgeons for not operating ! Retired surgeons, who are no longer active in the OR can provide lots of useful insights !
Why are doctors so cruel ?


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Nazis discovered in white supremacist military group?
Four members of one of the world's most dangerous white supremacist terrorist organisations have been arrested and accused of being members of a banned neo-Nazi terrorist group.
Four men have been arrested under suspicion of being members of proscribed neo-Nazi youth group National Action (NA). All four are serving soldiers in the British Army according to the Ministry of Defence.
Anti-fascists are largely unsurprised that neo-Nazis would consider joining the British Army, an organisation with one of the worst track records of racist violence in the entire history of humanity.
The four men arrested by West Midlands police are a a 22-year-old from Birmingham, a 32-year-old from Powys, mid Wales, a 24-year-old from Ipswich and a 24-year-old from Northampton. The four are being held in the West Midlands
In a statement, police said the suspects were arrested "on suspicion of being a member of a proscribed organisation (National Action) contrary to sec 11 of the Terrorism Act". Being a member of NA can lead to 10 years in jail.
It is relatively common for members of extreme right-wing organisations to join their respective nations' militaries. Earlier this year inspections were ordered at every German army barracks after Nazi memorabillia was found at two of them.
Sep 5 2017 15:06
Fozzie
"one of the world's most dangerous white supremacist terrorist organisations"
LOL. This is giving them more credit than they deserve, surely?
Mike Harman
Fozzie wrote:
That's pretty accurate for the British Army isn't it?
Ah! Duh. Apologies!
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redpajama_set_name: RedPajamaCommonCrawl


text:
Ammoland
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West Virginia State Police & Wyoming Highway Patrol Upgrade to New Smith & Wesson Pistols
Ammoland Inc. Posted on August 16, 2011 by Ammoland
Smith & Wesson M&P40 Polymer Pistols
SPRINGFIELD, Mass.--(Ammoland.com)- Smith & Wesson Holding Corporation (NASDAQ: SWHC), parent company of Smith & Wesson Corp., the legendary 159-year old company in the global business of safety, security, protection and sport, announced today that the West Virginia State Police and the Wyoming Highway Patrol recently chose to upgrade their current department-issued side arms with new pistols from Smith & Wesson.
The West Virginia State Police selected the Smith & Wesson Model 4566 TSW pistol and has received 750 pistols chambered in .45 ACP. The Wyoming Highway Patrol selected the Smith & Wesson M&P40 pistol and is currently in the process of transitioning its officers over to the new duty firearm.
West Virginia State Police
After evaluating competitive firearms, officials within the West Virginia State Police Department chose to continue using the Smith & Wesson 3rd Generation Model 4566 TSW metal frame pistols. The new firearms ordered by the department feature two new enhancements over the previously issued side arms. As requested by the department, the new pistols feature a Picatinny-style equipment rail for optional accessories and a Melonite® finish for increased durability. With the deployment of the new pistol, Department officers are currently using Smith & Wesson products for both their pistol and their patrol rifle duty firearms.
The West Virginia State Police issued the following statement regarding the new firearms:
"The West Virginia State Police Department has selected the Smith & Wesson 4566 TSW pistols in addition to the recent procurement of the Smith and Wesson M&P15 patrol rifles. Smith & Wesson's Law Enforcement support team has been exceptional in the understanding and insight provided for the selection of these weapons for the West Virginia State Troopers. The West Virginia State Police expects to maintain this professional relationship with Smith & Wesson to ensure that its troopers are adequately equipped to help ensure the safety of the public and the law enforcement officers themselves."
Wyoming Highway Patrol
The Wyoming Highway Patrol has received 237 M&P40 pistols that replace currently issued, firearms supplied by a European manufacturer. Officials at the Wyoming Highway Patrol indicated that they selected the .40 S&W polymer-frame pistol because of its enhanced ergonomics, accuracy and the ability to customize the firearms to each individual officer's preference.
James Debney, President of Firearms for Smith & Wesson, said, "We are pleased to continue our longstanding relationship with the men and women of the West Virginia State Police and we are pleased to begin a new relationship with the officers at the Wyoming Highway Patrol. Together, we have concentrated our efforts on supplying these first responders with the high-quality, professional products on which law enforcement officials have come to rely from Smith & Wesson. Our M&P40 and 4566 TSW pistols, along with our M&P15 tactical rifles, were engineered to meet the exact specifications of each department and all three models adhere to the strict qualifications we place on every Smith & Wesson firearm in terms of quality, durability and accuracy."
About Smith & Wesson
Smith & Wesson Holding Corporation (NASDAQ: SWHC) is a U.S.-based, global provider of products and services for safety, security, protection, and sport. The company delivers a broad portfolio of firearms and related training to the military, law enforcement, and sports markets, and designs and constructs facility perimeter security solutions for military and commercial applications. Smith & Wesson companies include Smith & Wesson Corp., the globally recognized manufacturer of quality firearms; Universal Safety Response, Inc., a full-service perimeter security integrator, barrier manufacturer, and installer; and Thompson/Center Arms Company, Inc., a premier designer and manufacturer of premium hunting firearms. Smith & Wesson facilities are located in Massachusetts, Maine, and Tennessee. For more information on Smith & Wesson and its companies, call (800) 331-0852 or log on to www.smith-wesson.com; www.usrgrab.com; or www.tcarms.com.
Cam in Bedford
Seems like an add. I now the title "upgrade". What specifically is better than the gun they are replacing? Why is this a upgrade and not simply a replacement?
kajunkkrittter
the M&P 45 is the only pistol with fixed sight that i own and hit where i aim
have also a 4506 and 1006 these are call 3rd generation pistols and i love um these are larger frame guns and are heavy
now i have 3 1911s in 45 and 10mm
im not afraid to carry any of them
S+W makes great revolvers but I was never a fan of their pistols. I used to own a 4506 and it was horrible. I'd like to hear some field reviews of this new pistol.
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It is modernly debated whether application of the free will has potential to cause harm to nature. Power possessed to the discourse, sensory/perceptual, physical influences on life experience by the slow moving machinery of change is a viral element in the problems of civilization; failed resolution of historical paradox involving mind and matter is a recurring source of problems. Reference is taken from the writing of Euclid in which a oneness of nature as an indivisible point of thought is made prerequisite in criteria of interpretation to demonstrate that contemporary scientific methodologies alternately ensue from the point of empirically centered induction. A qualification for conceptualizations is proposed that involves a physically describable form bound to energy in addition to contemporary notions of energy bound to form and a visually based mathematical-physical form is elaborated and discussed with respect to biological and natural processes.


text:
East Anglian Daily Times > Sport > Ipswich Town FC
Gill on Northampton clash, 'green shoots' and squad rotation
Published: 10:56 AM February 15, 2021
Matt Gill is looking for a new senior coaching role after being let go by Ipswich Town. Photo: Ross Halls - Credit: Archant
Ipswich Town first team coach Matt Gill hopes the 'green shoots' from the recent 2-0 home win against Blackpool continue to grow in tomorrow night's League One clash with Northampton Town at Portman Road (7pm).
The Blues produced a much-improved performance in victory over the Tangerines nine days ago, but soon followed that up with a disappointing 2-1 midweek loss at Peterborough United and then saw Saturday's scheduled match at in-form Shrewsbury postponed less than an hour before kick-off due to a partly frozen pitch.
Second-bottom Northampton arrive in Suffolk having lost eight of their previous 12 league matches, Keith Curle having recently been sacked and Jon Brady currently in caretaker charge.
"Peterborough was disappointing, very disappointing," said Gill. "There was the green shoots of the battling performance away at Crewe (1-1), then a very decent performance at home to Blackpool, so it felt disappointing last Tuesday night.
"I'm not particularly someone who wants to dwell. You have to learn from defeats and try and improve. That's something we need to take into tomorrow. Hopefully those green shoots from Blackpool can continue to grow.
"Obviously we're looking for a really positive performance. Northampton recently sacked their manager, they're fighting relegation and they'll be desperately trying to get themselves out of the situation they're in. So it will be a tough game, but it's one where we need to show the initiative and be really positive."
Northampton lost 2-0 at home to rock-bottom Burton on Saturday, conceding two late goals after seeing a goal chalked off for a marginal offside and also hitting the crossbar in the first half. The Cobblers switched to a back four under new boss Brady and were far less direct than previously under Curle.
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10 'A good team and well-coached' - Bolton boss Evatt on Town
"That's pretty much it in a nutshell," said Gill. "They obviously were quite direct under Keith and had unbelievable success last year in getting promoted during a real tough spell for the club, so his work shouldn't go unnoticed, but, yes, they did try to pass the ball a little bit more than they had done previously with the guy in charge at the moment.
"But I would like to think the majority of the work we do is on us and how we approach the game and how we can take the initiative."
On preparations having changed due to Saturday's postponement, Gill explained: "We'd have recovered on Sunday and some of the lads would have had a day off. Instead we came in yesterday (Sunday) and trained and had a real good session to make up for Saturday's miss and we'll train again today as per matchday minus one, as we call it, preparing for the game."
Asked if will be the same starting XI that had been picked to face Shrewsbury, Gill said: "Erm, it could be! Obviously we've got a lot of competition for places, as you know. Obviously it's three days later than Saturday would have been, so that brings other players into the fray. You'll have to wait and see."
Town boss Paul Lambert used a 4-3-3 formation all season up until last week's Peterborough game, for which he switched to a 4-4-2 diamond system. Could that be deployed again?
"I think with the times we're in you have to be adaptable and you have to be flexible," said Gill. "I think it's certainly an option. We went to it at Crewe late on in the game. Obviously we played a couple of different formations last year. I don't think you can rule anything out at this stage. You need to be flexible and you need the ability to adapt."
Striker Kayden Jackson is available again after serving a three-game ban for his red card against Sunderland.
"Kayden's been working hard, the same as a number of the lads," said Gill.
Asked if Jackson could be suited to playing in a front two, Gill replied: "I think Kayden's played in a two previously, both here and at previous clubs. It's an option. We've got quite a few options now, which is good.
"Players have the ability to play in a few different positions and a few different formations which I think is a plus for us."
And what about the potential to move captain Luke Chambers inside to play centre-half following the arrival of right-back Luke Matheson on loan from Wolves?
"It's obviously an option," said Gill. "He's obviously played a lot of career at centre-half, so it's certainly an option."
With five games having now been postponed in what was already a compacted season, Ipswich now have 21 games to play in 84 days. As a result, Town boss Lambert has made it clear that squad rotation may have to return to manage a relentless Saturday-Tuesday schedule.
That was evident when the four changes were made to the team which started the 2-0 win against Blackpool for the following game against Peterborough.
Asked why players aren't capable of churning out the games, Gill replied: "That's probably a question for them! It's an interesting one. I think some players would like to play loads of football. I think some players get fatigued and you have to look after them.
"Obviously physical performance is a major part of football these days with the distances they are travelling, the sprint distances and the high speed running. There have been lots of injuries this year, even at the top level, so it's about trying to be wary of that and not put people in physical danger.
"Most of the lads are pretty robust though and are willing to play when they are needed."
Asked if players are ever told that they were being rested for one game but assured they'd be starting the next, Gill said: "There's a little bit of that. A little bit. It's one of them ones, who's physically ready to play again? I think it can be a dangerous thing having an eye on two or three games at a time.
"We've made it pretty clear that if you get in the team and do well there will be occasions where you need to come out for recovery purposes or just to make sure we are looking after you physically.
"It's a bit of everything really. You need to make sure you are prepared to roll out the same team if they've done well, but there will be occasions where you do need to make changes.
"I think having some strength in depth will certainly help us and it's something we need to take advantage of it needed."
Ipswich have dropped to 12th in the table but have at least one game in hand on nine of the teams above them. They are 10 points adrift of second, nine points adrift of fifth and five points adrift of sixth.
"My view is you have to win the next game," said Gill. "Ideally we'd be 15 points clear the top, but we're not. You can only affect what happens next. That's the view I've tried to take my whole career really. Take responsibility and affect what happens next. That's what we're trying to do. And the next game is Northampton. We need to be really positive going into that.
"There's obviously lots of football left to play. There were some strange results at the weekend, if you want to put it like that. There is a long way to go and I think there will be loads of twists and turns between now and the end of the season.
"Look at the amazing run Oxford have been on. (nine wins from 10). That just shows you it can be done.
"We've had a similar run at the start of the season when it was really positive, so we know we are capable of it.
"It's about doing everything we possibly can to get on another run like that."


After a leak earlier this year, HSBC has given the first public hearing surrounding its digital-only banking initiative known as Project Icebergs. A leak late last year revealed that the bank was reportedly building a stand-alone digital bank, similar to NAB's UBank, specifically for business customers in order to fend off competition from challenger banks. The group managing director and group chief operating officer for HSBC, Andy Maguire, told delegates at the Money 20/20 conference that HSBC wanted to offer innovation for its customers.
"HSBC is a commercial bank, that's what we started as. We really wanted to re-imagine how we work but not in a way that was unrecognisable," he said. Mr Maguire said that commercial banking was complicated and the bank did not want to just offer a simplified digital bank solution. "Commercial banking is complicated and you can't just make everything the way you like. We wish you could but we have to work within the rules," he said.
The rest of this article can be found at investordaily.com.au.
The Reserve Bank of Australia has stated that climate change presents significant risks and opportunities for Australia's economy. Deputy governor of the RBA Guy Debelle said that, traditionally, climate change was only seen to impact agriculture, but it was no longer the case. "Agriculture is the prism through which we have historically thought about the effect of climate on the economy.
"Today, climate change presents significant risks and opportunities for a broader part of the economy than agriculture, though the impact on agriculture continues to be significant," he said. Mr Debelle said that there was bound to be significant volatility around the outcome of climate change and the economy would need to adapt to that. "The trend changes [for climate change] aren't likely to be smooth. There is likely to be volatility around the trend, with the potential for damaging outcomes from spikes above the trend," he said.
The corporate regulator will have more powers to hold financial services industries to account for misconduct under a new government proposal. The consultation is in response to Recommendation 1.15 of the Hayne commission's final report to enhance the current approved codes framework in the Corporations Act 2001, Treasurer Josh Frydenberg said in a statement. The report noted that "there must be adequate means to identify, correct and prevent systemic failures in applying the code", and that "in order to do that, some provisions of the codes should be picked up and applied as law".
ASIC to take into account whether particular provisions of an industry code have been designated as enforceable code provisions in determining whether to approve a code.
Mr Frydenberg said that through these changes, it will also be made clear that certain promises made in codes are enforceable against financial services firms by consumers.
Virtual banking and open data might sound like mere buzzwords in most parts of the world, but in Hong Kong, they are becoming a reality. Over the past few years, Hong Kong has embarked on a journey of digital transformation in the financial services sector to safeguard its role as a leading global financial centre. It is a trendsetter and an example for other cities in the region to follow, especially when it comes to government support in creating the right regulatory ecosystem for digital transformation. But the transformation journey is not without risk.
The financial services industry accounted for a fifth of Hong Kong's economy in 2017. To transform into a digital financial centre, financial services players in the city must fully embrace fintech, not only to effectively lower costs, but also to achieve better customer satisfaction. As fintech has great potential to enhance financial sector productivity and, consequently, boost the economy, private investment has been pouring into this space since 2015, growing annually at 127 percent during 2015-2017—higher than the global average. Investments have been made in payments and lending, as well as in other areas, such as data and security, remittances and insurance, thereby creating a more balanced ecosystem.
The rest of this article can be found at brinknews.com.
Utilising technology to provide simple, easy-to-use, and accessible apps is paramount to appeal to young investors, according to data and analytics company GlobalData. GlobalData's '2018 Mass Affluent Investors Survey' found that 53.3 per cent of mass affluent investors between the ages of 18–34 agree they are easily flustered when things are complicated. The company sees this as proof that products should be kept simple in order to reach the young mass affluent population and encourage them to begin investing.
"It is important to remember that while these apps may seem easy to use to the well-informed, tech-savvy crowd, it is just as important to ensure the same is true for less technologically advanced individuals, or even people who do not understand the market as much," GlobalData's associate wealth management analyst Oliver Wintle said. "Those providers that are not acute to this will lose out to competitors that provide platforms that resonate better with the masses." A digital product that is simple and easy to use can help companies welcome new users that were previously put off by jargon and complicated procedures, he added.


text:
\section{Introduction}
Major advances in wireless networking technologies enabling ultra-high data rates with low latency have led to the proliferation of computation intensive applications, such as augmented reality, interactive gaming and video streaming \cite{YLIU}. However, considering the exacting computation and storage requirements of these applications, the rate
of technological advancements on the device side has been generally slower, so that they can hardly meet such demands. Cloud computing, in which user devices offload their computation-intensive tasks to much more powerful remote servers, can help devices reduce their computation burden \cite{PPierleoni}. However, due to network congestion, cloud computing may not be suitable for real-time applications requiring ultra-low latency and very high bandwidth \cite{KKaur}. Mobile edge computing (MEC) provides an alternative to cloud computing by offloading computation to servers at the edge of the network, thereby reducing end-to-end delays and data processing bottlenecks \cite{XHE}. However, the computation capability of MEC servers is limited, and in situations involving extensive data traffic and high user density, some mobile devices might not be able to access them, which negatively impacts the quality of service \cite{JLI}.

As a complement to cloud and edge computing, fog computing provides a decentralized framework whereby the available computation resources of nearby mobile devices are exploited for task offloading through incentive policies \cite{MZENG,SLUO,RBeraldi}. Due to the proximity of the available resources, this type of task partitioning mechanism reduces the latency and the backbone traffic in the network, and in turn, increases energy efficiency \cite{YLAN}. Device-to-device (D2D) communication, which will play an important role in fifth generation (5G) and beyond 5G (B5G) networks, is a strong candidate to enable fog computing. In \cite{YLAN}, the authors focus on  maximization of total utility in terms of energy and time consumption in a D2D-aided fog computing scenario. In \cite{SJosilo}, minimization of average task completion time is considered by using a game theoretical model. Reference \cite{LPu} considers network-assisted D2D fog computing, in which the objective is to minimize time-average energy consumption, whereas in \cite{HXing} the problem of computation latency minimization in D2D fog computing is studied by considering an optimal task assignment strategy. However, the aforementioned studies do not consider CPU throttling during task offloading, as discussed below.

In mobile devices, dynamic thermal management (DTM) schemes control the on-chip temperature by lowering the voltage and frequency of the CPU to prevent damage in the case of high temperature \cite{JMKim}. Ideally, devices allocate the highest available CPU frequency, measured in cycles per second, to perform a given task within a minimum amount of time. Due to DTM, however, significant yet unpredictable fluctuations in allocated CPU frequency do occur \cite{OSahin}. Since real-time applications require low latency and strict processing times, a random reduction in CPU frequency negatively impacts task offloading. Consequently, to optimize fog computing performance subject to this type of uncertainty, allocation of computation resources should be treated as a probabilistic optimization problem rather than a deterministic one.

Motivated by the aforementioned challenges, we address the problem of optimal resource allocation in terms of  task  partitioning,  computation  resources  and transmit powers, in D2D-aided mobile fog computing. Specifically, we aim to minimize the expected total energy consumption under probabilistic constraints on the task processing time. Since the formulated problem is non-convex, finding the global optimum is generally intractable; therefore, we  propose  two  sub-optimal solution methods. The  first  method  leverages the difference of convex (DC) optimization framework combined with chance-constraint  programming  to  handle  the  probabilistic constraints. Nonetheless, we find that the performance of DC programming remains sensitive to the choice of the initial  point. To overcome this difficulty, we develop a second method that relies solely on convex programming. Simulation results demonstrate that the second method outperforms the first one in  terms  of  energy efficiency and  run-time, while both methods offer significant energy savings over local computation.

The paper is organized as follows. In Section II, we describe the system model and formulate the problem statement. In Section III, we present the proposed DC and convex programming methods. Simulation results are presented and discussed in Section IV. Finally, Section V concludes the paper.

\vspace{-.11cm}
\section{System Model and Problem Statement} \label{Sec:SystemModel}
\vspace{-.11cm}
\subsection{System Model}
\vspace{-.12cm}
We consider a wireless sub-network comprised of a single active device that has a computation-intensive task to perform and $J$ offloading devices which can be used to offload this task, as seen in Fig. \ref{Network}. We label the active device by $0$ and the offloading devices by
$j \in \mathcal{J} = \{1,2,\ldots\}$.
Similar to \cite{SJosilo}, we assume  simultaneous orthogonal side-links to establish D2D communications between the active device and each one of the offloading devices prior to task offloading.
\begin{figure}[h]
	\centering
	\includegraphics[width=60mm]{Figs/Network.eps}
	\small
	\caption{D2D-aided fog computing scenario, where an active device (indexed by  $0$) can offload its tasks to nearby offloading devices (indexed by $j\in \mathcal{J}$) .}
	\normalsize
	\label{Network}
\end{figure}

The computation task  of the active device is characterized by the tuple $( b,c,t^{\text{max}})$.
Here, $b$ indicates the task size in bits, $c$ denotes the number of CPU cycles required to process one bit of data, and $t^{\text{max}}$ is the maximum time limit for completing the task.
The device may compute its task locally  and/or partially offload it to one or more offloading devices in $\mathcal{J}$.
Accordingly, the task size can be decomposed as:
\vspace{-.02cm}
\begin{align}
 b=b_{0} + \sum_{j\in\mathcal{J}}{b_{j}} \label{DataArranged}
\end{align}
where $b_{0}$ and $b_{j}$ represent the portions kept at the active device and sent to the $j$th offloading device, respectively.
These portions are collected in the vector $\boldsymbol{b}=[b_{0}\ b_{1}\hspace{-0.1cm}\ \ldots \hspace{-0.1cm}\ b_{J}]^\top$ where ${}^\top$ denotes the transpose operation.

To compute the local portion of its task, the active device allocates a part $f_{0}$ of its computation resources, measured in CPU cycles per second, which cannot exceed its  maximum computation capability $ f_0^{\max}$.
However, due to unpredictable CPU throttling, e.g., resulting from temperature fluctuations, the actual computation resource used by the device is $\Tilde{f}_{0}=(1-{\xi}_0)f_{0}$, where ${\xi}_0\in[0,1]$ is a random variable with known distribution. Denoting the time it takes to complete the local portion of the task at the active device as $t^{\text{co}}_0$, we can write:
\begin{align}
\Tilde{f}_{0}t^{\text{co}}_0= b_{0}c \label{LocalCompTime}
\end{align}
The energy consumed for local computation is given by \cite{NLi}:
\begin{align}
E^{\text{loc}}=\kappa b_{0} c \Tilde{f}_{0}^2 = \kappa \Tilde{f}_{0}^3 t^{\text{co}}_0\label{LocalEnergy}
\end{align}
where $\kappa$ is an effective capacitance constant that depends on the chip architecture.

The active device also uploads to the $j$th offloading device the corresponding task portion of size $b_j$. The achievable data rate for transmission to the $j$th device is:\\*
\vspace{-.1cm}
\begin{align}
R_{j}=\hspace{.5mm} W\mbox{log}_2\bigg(1+\dfrac{ P_{j} G_{j}}{ N_0 }\bigg)
\label{ChannelCapacity}
\end{align}
where $P_{j}$ is the allocated transmission power, $G_{j}$ is the channel gain, $W$ is the channel bandwidth, and $ N_0$ is the noise power. Denoting by $t^{\text{up}}_{j}$ the upload time, we have:
\vspace{-.1cm}
\begin{align}
b_{j}=R_{j}t^{\text{up}}_{j}   \label{TaskUpload}
\end{align}
\vspace{-.55cm}

As in the case of the active device, the $j$th offloading device allocates a part $f_{j}$ of its computation resources, which cannot exceed its maximum computation capability $ f_{j}^{\max}$, to complete the offloaded task. As before, the actual computation resource used is $\Tilde{f}_{j}=(1-{\xi}_j)f_{j}$ where ${\xi}_j$ is a random variable with known distribution.
Then, similar to (\ref{LocalCompTime}), we have:
\vspace{-.1cm}
\begin{align}
 \Tilde{f}_{j}t^{\text{co}}_{j} = b_{j}c\label{TaskCompute}
\end{align}
where $t^{\text{co}}_{j}$\hspace{-.04cm} is the time it takes the complete the offloaded portion.

Overall, the energy consumed to upload and compute the offloaded tasks is:
\vspace{-.16cm}
\begin{align}
E^{\text{off}}&=\sum_{j\in\mathcal{J}} \big(P_{j}t^{\text{up}}_{j}+\kappa \Tilde{f}_{j}^3t^{\text{co}}_{j}\big)\label{OffloadingEnergy}
\end{align}
Finally, the total energy consumed to complete the task can be given as a sum of two terms:
\begin{align}
E&= E^{\text{loc}} + E^{\text{off}}
=\underbrace{\sum_{j\in\mathcal{J}}\frac{P_{j}b_{j}}{R_{j}}}_{\phi(\textbf{p},\textbf{b})} + \underbrace{\sum_{i\in\mathcal{I}}\kappa b_{i} c \Tilde{f}_{i}^2}_{\psi(\textbf{b},\Tilde{\textbf{f}})}
\label{TotalEnergy}
\end{align}
where $\mathcal{I}=\{0\}\cup \mathcal{J}$, while $\phi(\textbf{p},\textbf{b})$ and $\psi(\textbf{b},\Tilde{\textbf{f}})$ are the total task uploading energy and the total task computation energy, respectively. Furthermore, $\textbf{p}=[ P_{1}\  ...\ P_{J}]^\top$ contains the transmit powers of the active device to its offloading devices and $\Tilde{\textbf{f}}=[\Tilde{f}_{0}\ \Tilde{f}_{1}\ ...\ \Tilde{f}_{J}]^\top$ contains the actual computation resources used by the devices.

\subsection{Problem Statement}
In this paper, we address the problem of optimal resource allocation, in terms of task portions, computation resources and transmit powers, in the above D2D-aided fog computing scenario. Since the allocated computation resources have a random nature, we aim to minimize the expected value of the total energy consumption subject to probabilistic constraints on the task processing times:
\vspace{-.1cm}
\begin{subequations}
\begin{align}
\mathscr{P}_1:  \min_{\textbf{p}, \textbf{b}, \textbf{f}, \textbf{t}} \quad \mathbb{E}_{\boldsymbol{\xi}}\big[E\big] \label{Prob1}\\
&\hspace{-2.72cm}\textrm{s.t.} \quad  0 \le \sum_{j\in\mathcal{J}}P_{j}\le P^\text{max}\label{C1}\\
&\hspace{-1.93cm} \sum_{i\in\mathcal{I}}{b_{i}} =b\label{C2} \\
&\hspace{-1.93cm}  \mathbb{P}(t^{\text{co}}_0\le t^{\text{max}})\ge \gamma  \label{C3} \\
&\hspace{-1.93cm}  \mathbb{P}\big( t^{\text{co}}_{j}\le\hspace{-.8mm} (t^{\text{max}}-t^{\text{up}}_{j})\big)\hspace{-.8mm}\ge \gamma , \forall j\hspace{-.8mm}\in\hspace{-.8mm}\mathcal{J} \label{C4} \\
&\hspace{-1.93cm}  b_{j}=R_{j}t^{\text{up}}_{j},\ \forall j\in\mathcal{J}\label{C5} \\
&\hspace{-1.93cm}  f_{0}\le f^{\text{max}}_0, f_{j}\le f^{\text{max}}_j, \forall j\in\mathcal{J} \label{C7} \\
\vspace*{-.1cm}
&\hspace{-1.93cm} {b}_0, b_{j}, f_{0}, f_{j}, t^{\text{up}}_{j}\ge0 \quad  \forall j\in\mathcal{J}\label{C8}
\end{align}
\end{subequations}
where $\textbf{f}=[f_{0}\ f_{1}\ ...\ f_{J}]^\top$ contains the allocated computation resources and $\textbf{t}=[t^{\text{up}}_{1}\ ...\ t^{\text{up}}_{J}]^\top$ includes the task upload times to each offloading devices. Also, $\mathbb{E}_{\boldsymbol{\xi}}[\cdot]$ is the expectation operator, and $\mathbb{P}(\cdot)$ is the probability of an event.

In problem $\mathscr{P}_1$, the expectation in (\ref{Prob1}) is taken over the distribution of the random vector $\boldsymbol{\xi} = [\xi_{0}\ \xi_{1}\ ...\ \xi_{J}]^\top$, constraint (\ref{C1}) limits the total transmit power of the active device to $P^\text{max}$ while constraint (\ref{C2}) guarantees that the task portion sizes add up to the original task size. Constraints (\ref{C3}) and (\ref{C4}) stipulate that the probability of completing the task within the maximum time limit $t^{\text{max}}$ is higher than a given reliability level $\gamma \in [0,1]$.
Constraint (\ref{C5}) ensures that the channel rate and corresponding task uploading time are consistent with the allocated task portions. Finally, constraint (\ref{C7}) indicates that the allocated computation resources cannot exceed the computation capabilities of the devices and constraint (\ref{C8}) expresses the non-negative nature of the decision variables.

\section{Proposed Sub-Optimal Methods} \label{Sec:SubOptTaskOff}
Due to the non-convex objective function (\ref{Prob1}) and the non-convex constraints (\ref{C3}), (\ref{C4}), problem $\mathscr{P}_1$ is intractable. Therefore, in this section, we propose two sub-optimal methods to solve problem $\mathscr{P}_1$.

In the first method, we write the non-convex objective function and the non-convex constraints as difference of convex functions (DCF), while using chance-constraint programming to handle the probabilistic time constraints.
The new optimization problem can then be solved using DC programming. In the second method, to address certain issues related to initialization of the DC programming-based method, we propose a more effective two-step approach which relies solely on convex programming.

\subsection{DC Programming Method}
We start by writing the task uploading energy
$\phi(\textbf{p},\textbf{b})$ introduced in (\ref{TotalEnergy}) as a DCF:
\begin{align}
\phi(\textbf{p},\textbf{b})&=\phi_1(\textbf{p},\textbf{b}) - \phi_2(\textbf{p},\textbf{b}) \label{UploadingEnergy}
\end{align}
where $\phi_1(\textbf{p},\textbf{b})=\sum_{j\in\mathcal{J}}(P_{j}+\frac{b_{j}}{2R_{j}})^2$ and $\phi_2(\textbf{p},\textbf{b})=\sum_{j\in\mathcal{J}}( P_{j}^2 + \frac{b^2_{j}}{4R^2_{j}})$. We also decompose the expected value of the total computation energy $\psi(\textbf{b},\tilde{\textbf{f}})$ as follows:
\begin{align}
\mathbb{E}_{\boldsymbol{\xi}}[\psi(\textbf{b},\tilde{\textbf{f}})]&=\mathbb{E}_{\boldsymbol{\xi}}\bigg[ \sum_{i\in\mathcal{I}}\kappa b_{i} c (1-\xi_i)^2f_{i}^2\bigg]\nonumber\\
&=\kappa c  \sum_{i\in\mathcal{I}}\eta_i \big( (b_{i} + f_{i}/2)^2 - (b^2_{i} + f^2_{i}/4) \big)\nonumber\\
&=\psi_1(\textbf{b},\textbf{f})-\psi_2(\textbf{b},\textbf{f})\label{ExpectationComptEnergy}
\end{align}
 where $\psi_1(\textbf{b},\textbf{f})= \kappa c  \sum_{i\in\mathcal{I}}\eta_i  (b_{i} + f_{i}/2)^2$, $\psi_2(\textbf{b},\textbf{f})= \kappa c   \sum_{i\in\mathcal{I}}\eta_i  (b^2_{i} + f^2_{i}/4)$ and $\eta_i=\mathbb{E}[(1-\xi_i)^2],\ i\in \mathcal{I}$. Finally, the objective function (\ref{Prob1}) expressed as a DCF:
 \begin{align}
\mathbb{E}_{\boldsymbol{\xi}}[E]&\equiv H(\textbf{x})=Y(\textbf{x})-Z(\textbf{x})\label{DC_objective}
\end{align}
where $Y(\textbf{x})=\phi_1(\textbf{p},\textbf{b}) +\psi_1(\textbf{b},\textbf{f})$ and $Z(\textbf{x})=\phi_2(\textbf{p},\textbf{b}) +\psi_2(\textbf{b},\textbf{f})$ are convex functions, and   $\textbf{x}=[\textbf{p}^\top  \textbf{b}^\top \textbf{f}^\top \textbf{t}^\top ]^\top$  contains all the search variables for convenience.

As shown in \cite{AlexanderS}, to apply a DC algorithm, each non-convex equality and inequality constraints can be incorporated into (\ref{DC_objective}) by using a penalty parameter once their DCFs are available. However, for our problem, we found that this approach yielded slow convergence. Whereas in \cite{AhmadiAA}, a DC algorithm is applied to a problem consisting of  only non-convex inequality constraints that are decomposed as DCFs. Hence, if we eliminate the equality constraint (\ref{C5}) by incorporating it into (\ref{DC_objective}) based on the penalty approach in \cite{AlexanderS}, we can develop a DC-based algorithm as in \cite{AhmadiAA} to solve a problem involving a penalized objective function (which is shown to be DCF \cite{AlexanderS}) with only inequality constraints (\ref{C3}) and (\ref{C4}).

To this end, we decompose the non-convex equality constraint (\ref{C5}) as:
\begin{align}
C^{\text{eq}}_j(\textbf{x})&=\dfrac{b_{j}}{R_{j}}-t^{\text{up}}_{j} =Y^{\text{eq}}_j(\textbf{x}) - Z^{\text{eq}}_j(\textbf{x}),\ \forall j\in\mathcal{J}\label{Det_C4_Decomposed_Final}
\end{align}
where $Y^{\text{eq}}_j(\textbf{x})=(b_{j}+ \frac{1}{2R_j})^2$ and $Z^{\text{eq}}_j(\textbf{x})=( b^2_{j} + \frac{1}{4R^2_j} + t^{\text{up}}_{j})$ are convex functions. Then we introduce the penalty term in the objective function, which can be also written as a DCF \cite{AlexanderS}:
\begin{align}
H_\lambda(\textbf{x})&=Y_\lambda(\textbf{x})-Z_\lambda(\textbf{x}) \label{PenalizedObj}
\end{align}
where
\begin{align}
Y_\lambda(\textbf{x})\hspace{-.07cm}&=\hspace{-.07cm}Y(\textbf{x})\hspace{-.07cm} +\hspace{-.07cm} 2\lambda\sum_{j\in\mathcal{J}}\max\big\{\hspace{-.07cm} Y^{\text{eq}}_j(\textbf{x});Z^{\text{eq}}_j(\textbf{x})\big\}\\
Z_\lambda(\textbf{x})&=Z(\textbf{x}) + \lambda\sum_{j\in\mathcal{J}} (Y^{\text{eq}}_j(\textbf{x})+Z^{\text{eq}}_j(\textbf{x}))
\end{align}
and $\lambda\ge 0$ is the penalty parameter.

In order to deal with the probabilistic inequality constraints (\ref{C3}) and (\ref{C4}), we adopt the chance-constraint programming approach \cite{Charnes}, and transform them into their deterministic equivalents. Specifically, constraint (\ref{C3}) can be given in terms of the cumulative distribution function (CDF) of $\xi_0$, $\textit{F}_{\xi_0}(\cdot)$, as follows:
\begin{align}
\mathbb{P}\bigg(\dfrac{b_{0}c}{(1-{\xi}_0)f_{0}}\hspace{-.06cm}\le\hspace{-.06cm} t^{\text{max}}\hspace{-.06cm}\bigg)\hspace{-.06cm}&=\hspace{-.06cm}\mathbb{P}\bigg(\hspace{-.05cm}{\xi}_0\hspace{-.06cm}\le \hspace{-.06cm}\underbrace{ \dfrac{f_{0}t^{\text{max}}-b_{0}c}{f_{0}t^{\text{max}}}}_{z}\bigg)\hspace{-.06cm}=\textit{F}_{\xi_0}(z)\ge  \gamma \nonumber
\end{align}
Then, assuming $\textit{F}_{\xi_0}(\cdot)$ is invertible, we can obtain the deterministic form of constraint (\ref{C3}) as follows:
\begin{align}
C_0(\textbf{x})=\dfrac{f_{0}t^{\text{max}}-b_{0}c}{f_{0}t^{\text{max}}}-F^{-1}_{\xi_0}(\gamma)\ge 0 \label{CDF_C3}
\end{align}
\noindent
where $\textit{F}^{-1}_{\xi_0}(\gamma)$ is the inverse CDF evaluated at $\gamma$. The new deterministic constraint (\ref{CDF_C3}) can now be written as a DCF in the following way:
\begin{align}
C_0(\textbf{x})=\text{ln}(\dfrac{b_{0}}{f_{0}}) - \text{ln}(q_0)=Y_0(\textbf{x}) - Z_0(\textbf{x})\le 0
\label{Det_C3}
\end{align}
\noindent
where  $q_0=t^{\text{max}}c^{-1}(1-F^{-1}_{\xi_0}(\gamma))$ is a non-negative constant, and $Y_0(\textbf{x})=-\text{ln}(f_0)$ and $Z_0(\textbf{x})=-\text{ln}(b_0) + \text{ln}(q_0)$ are convex functions.

Proceeding in a similar way, the deterministic form of constraint (\ref{C4}) is:
\begin{align}
C_j(\textbf{x}) =\dfrac{f_{j}(t^{\text{max}}-t^{\text{up}}_{j})-b_{j}c}{f_{j}(t^{\text{max}}-t^{\text{up}}_{j})}-F^{-1}_{\xi_j}(\gamma)\ge 0,\ \forall j\in\mathcal{J}
\label{Det_C4}
\end{align}
where $F^{-1}_{\xi_j}(\gamma)$ is the inverse CDF of $\xi_j$ evaluated at $\gamma$. In turn, (\ref{Det_C4}) can be decomposed as follows:
\begin{align}
\hspace{-0.2cm}C_j(\textbf{x})\hspace{-0.05cm}=\hspace{-0.05cm}\dfrac{t^{\text{up}}_{j}f_jq_j}{t^{\text{max}}}\hspace{-0.05cm}+\hspace{-0.05cm}b_{j}\hspace{-0.05cm}-\hspace{-0.05cm}f_jq_j\hspace{-0.05cm}=\hspace{-0.05cm}Y_j(\textbf{x})\hspace{-0.05cm}-\hspace{-0.05cm}Z_j(\textbf{x})\le 0,\hspace{-0.05cm}\ \forall j\hspace{-0.05cm}\in\hspace{-0.08cm}\mathcal{J}
\label{Decomposed_C4}
\end{align}
where  $q_j=t^{\text{max}}c^{-1}(1-F^{-1}_{\xi_j}(\gamma))\ \forall j\in\mathcal{J}$ is a non-negative constant, and $Y_j(\textbf{x})=\frac{q_j}{t^{\text{max}}}(
t^{\text{up}}_{j} + f_j/2)^2 + b_{j}$ and $Z_j(\textbf{x})=\frac{q_j}{t^{\text{max}}}\big((t^{\text{up}}_{j})^2 + f^2_j/4\big) + f_jq_j$ are convex functions.

Our first method is finally obtained by combining the DC programming approach in \cite{AhmadiAA} with the penalized DC approach in \cite{AlexanderS}; the resulting procedure is presented as Algorithm \ref{Iterative_PowerReducetion}. Following initialization, at the $k$th iteration of the algorithm, we first determine the convex approximations $H_\lambda^{(k)}(\textbf{x})$ and $C_i^{(k)}(\textbf{x})$ of $H_\lambda(\textbf{x})$ and  $C_i(\textbf{x})$ in step \ref{H_CONVEX} and \ref{C_CONVEX}, respectively, where $\nabla$ denotes the gradient operator. In step \ref{Solve_H}, we minimize $H_\lambda^{(k)}(\textbf{x})$ subject to the indicated constraints using standard convex optimization techniques until the sequence $\{H_\lambda^{(k)}(\textbf{x})\}$ converges with tolerance $\epsilon$ or the maximum iteration number $k^{\text{max}}$ is reached. The algorithm outputs the desired vector $\textbf{x}^{(k)}$ of the allocated resources.
 \begin{algorithm}[H]
	\caption{DC Algorithm Method}\label{Iterative_PowerReducetion}
	\begin{algorithmic}[1]
	\State $\textbf{input}$ Set $k=0$, initialize $\textbf{x}^{(0)}$
    \Repeat
    	\State \hspace{-.4cm}$H_\lambda^{(k)}(\textbf{x})=Y_\lambda(\textbf{x})-Z_\lambda(\textbf{x}^{(k)})-\nabla Z_\lambda(\textbf{x}^{(k)})^\top(\textbf{x}-\textbf{x}^{(k)})  $\label{H_CONVEX}
    	\State \hspace{-.4cm}$C_i^{(k)}(\textbf{x})\hspace{-.1cm}=\hspace{-.1cm}Y_i(\textbf{x})\hspace{-.07cm}-\hspace{-.07cm}Z_i(\textbf{x}^{(k)})\hspace{-.1cm}-\hspace{-.1cm}\nabla Z_i(\textbf{x}^{(k)})^\top(\textbf{x}\hspace{-.1cm}-\hspace{-.1cm}\textbf{x}^{(k)}),\ i \in \mathcal{I} $\label{C_CONVEX}
    	 \State \hspace{-.4cm}solve $\textbf{x}^{(k+1)}=\argmin\limits_{\textbf{x}}H_\lambda^{(k)}(\textbf{x})$ \label{Solve_H}
    	 \State \hspace{2.2cm} $ \textrm{s.t.}\  C_i^{(k)}(\textbf{x})\le 0,\ i\in \mathcal{I}$
    	  \State \hspace{2.7cm} $ (\ref{C1}), (\ref{C2}), (\ref{C7})\ \text{and}\ (\ref{C8})$
	 \State \hspace{-.4cm} $k\leftarrow k+1$
	\Until{$|H_\lambda(\textbf{x}^{(k+1)})-H_\lambda( \textbf{x}^{(k)})|>\epsilon$ \textbf{or} $k\le k^{\text{max}}$}
    \State $\textbf{output}\ \textbf{x}^{(k)}$
	\end{algorithmic}
\end{algorithm}
\noindent

\subsection{Convex-Programming Method}
Although DC programming guarantees a local optimum by converging to a stationary point \cite{Vucic}, its performance depends heavily on the choice of the initial point $\textbf{x}^{(0)}$. To address this limitation, we propose a more  effective  two-step  approach relying solely on convex programming, which eliminates the dependence on user-defined initialization.

Consider an ideal scenario, in which there is no uncertainty in the allocated computation resources and the task uploading is instantaneous, i.e., $\xi_i=0, \forall i\in \mathcal{I}$ and $t^{\text{up}}_{j}=0, \forall j\in \mathcal{J}$. For this scenario let $\textbf{f}^{\star}$ and $\textbf{b}^{\star}$ be the optimal computation resources and the optimal task partitioning subject to constraint (\ref{C2}), which gives the minimum total energy consumption as $E^\star$. Note that based on (\ref{LocalCompTime}) or (\ref{TaskCompute}), we have  $f^{\star}_{i}t^{\text{co}}_i= b^{\star}_{i}c, \forall i\in\mathcal{I}$. It can be seen that at the optimal solution, the task completion time must match the given deadline, i.e., $t^{\text{co}^\star}_i=t^{\text{max}}$, since there cannot be any other computation resources, say $f_{i}^+\ \forall i\in \mathcal{I}$ with $f_{i}^+<f_{i}^\star$ that can reduce further the total energy consumption $E^\star$ without violating the time constraint or constraint (\ref{C2}).

Based on the above, we can write the total computation energy in terms of only transmit power and task partitioning by replacing $t_0^{\text{co}}$ with $t^{\text{max}}$ and $t_{j}^{\text{co}}$ with $(t^{\text{max}}-t_{j}^{\text{up}})$ as follows:
\begin{align}
\psi(\textbf{p},\textbf{b})= \frac{\kappa (b_{0} c)^3 }{{(t^{\text{max}})}^2} + \sum_{j\in\mathcal{J}}  \frac{\kappa (b_{j} c)^3 }{\big(t^{\text{max}}-\frac{b_{j}}{R_{j}}\big)^2} \label{ComputationEnergy}
\end{align}
Hence, we decouple the allocation of computation resources and task partitioning in (\ref{ComputationEnergy}). More importantly, it can be shown that (\ref{ComputationEnergy}) is a convex function over the convex feasible set defined by constraints (\ref{C1}) and (\ref{C2}). Therefore, in the first step of our convex-programming method, we minimize the convex part $\psi(\textbf{p},\textbf{b})$ subject to constraints (\ref{C1}), (\ref{C2}) and a modified form of constraint (\ref{C5}) from problem $\mathscr{P}_1$:
\begin{subequations}
\begin{align}
\mathscr{P}_2:  \min_{\textbf{p}, \textbf{b}} \quad \psi(\textbf{p},\textbf{b}) \label{Prob2}\\
&\hspace{-2.5cm}\textrm{s.t.} \quad  0 \le \sum_{j\in\mathcal{J}}P_{j}\le P^\text{max} \label{P2_C1}\\
&\hspace{-1.73cm}   \sum_{i\in\mathcal{I}}{b_{i}} =b,\ b_i\ge0   \label{P2_C2}\\
&\hspace{-1.73cm} b_{j} - \alpha R_{j}t^{\text{max}}\le 0,  \forall j\in\mathcal{J} \label{P2_C3
\end{align}
\end{subequations}
Problem $\mathscr{P}_2$ can be easily solved by means of standard convex optimization methods. In constraint (\ref{P2_C3}), the scaling parameter $\alpha\in(0,1)$ is used to avoid the task uploading time exceeding the maximum time limit, i.e., $t^{\text{up}}_{j}>t^{\text{max}}$. In this way, constraint (\ref{P2_C3}) allows the computation time $t^{\text{co}}_{j}\ \forall j\in \mathcal{J}$ to be within the maximum time limit, and consequently, the solution of $\mathscr{P}_2$ lies in the feasible set of problem $\mathscr{P}_1$. We denote the solution of Problem $\mathscr{P}_2$ as $\textbf{b}^*$ and $\textbf{p}^*$.

In the second step, we minimize the expectation of the total computation energy (\ref{ExpectationComptEnergy}) with respect to computation resources subject to deterministic equivalents of constraints (\ref{C3}) and (\ref{C4}),  wherein the optimal values of $\textbf{b}^*$ and $\textbf{p}^*$ from Problem $\mathscr{P}_2$ are used in place of $\textbf{b}$ and $\textbf{p}$.
\\
\vspace{-.58cm}
\begin{subequations}
\begin{align}
\mathscr{P}_3:  \min_{\textbf{f}} \quad
\mathbb{E}_{\boldsymbol{\xi}}[\psi(\textbf{b}^*,\tilde{\textbf{f}})&]= \sum_{i\in\mathcal{I}}\kappa  c \eta_i b^*_{i} f_{i}^2\label{Prob3}\\
&\hspace{-2.4cm}\textrm{s.t.} \quad
\frac{b^*_{0}c}{t^{\text{max}}(1-F^{-1}_{\xi_0}(\gamma))}\le f_{0} \label{P3_C1} \\
&\hspace{-1.65cm}
\frac{b^*_{j}c}{(t^{\text{max}}-t^{\text{up}^*}_{j})(1-F^{-1}_{\xi_j}(\gamma))}\le f_{j}, \forall j\hspace{-.8mm}\in\hspace{-.8mm}\mathcal{J}\label{P3_C2}
\end{align}
\end{subequations}
Note that $t^{\text{up}^*}_{j}=\frac{b^*_{j}}{R^*_{j}}\ \forall j\in\mathcal{J},$  where $R^*_{j}$ is the corresponding data rate for $P_j^*$; we then form the vector $\textbf{t}^*$ accordingly. It can be seen that the optimal solution $\textbf{f}^*$ of problem $\mathscr{P}_3$ can be directly calculated since it satisfies constraints (\ref{P3_C1}) and (\ref{P3_C2}) with equality.

After solving problem $\mathscr{P}_3$, the allocated computation resource at an  offloading device, say $j$, might exceed its computation capability, i.e., $f^*_{j}> f^{\text{max}}_{j}$. In this case, the solution of $\mathscr{P}_3$ is not in the feasible set of $\mathscr{P}_1$ as constraint (\ref{C7}) is violated. To address this issue, we reduce $f^*_{j}$ to $ f^{\text{max}}_{j}$ and we adjust the initially allocated task portion $b^*_{j}$ so that it can be computed without violating constraint (\ref{C7}). Specifically, we replace $b^*_{j}$ by
\begin{align}
b^{+}_{j}=\dfrac{f^{\text{max}}_{j} R^*_{j}  t^{\text{max}} (1-F^{-1}_{\xi_j}(\gamma))}{R^*_{j}c+f^{\text{max}}_{j}(1-F^{-1}_{\xi_j}(\gamma))}\label{NewTaskSize}
\end{align}
which is  the maximum task portion size that can be computed by utilizing the full available computation resource $f^{\text{max}}_{j}$. The value of $b_{j}^+$
is obtained from constraint (\ref{P3_C2}) by replacing $f_{j}$ with $f^{\text{max}}_{j}$. Then the excess task portion, $b^{*}_{j}-b^{+}_{j}$, is assigned to the active device and/or the rest of the offloading devices. This is done by solving problem $\mathscr{P}_2$ and $\mathscr{P}_3$  after we remove the $j$th device from the set of available offloading destinations, i.e., we replace  $\mathcal{J}$ with $\mathcal{J}- \{j\}$. The process is repeated until constraint (\ref{C7}) is no longer violated by the remaining offloading devices.
If the set $\mathcal{J}$ becomes empty, then the leftover portion of the task size is computed at the active device, where we assume that $\frac{b^*_{0}c}{t^{\text{max}}(1-F^{-1}_{\xi_j}(\gamma))}= f^*_{0}< f^{\text{max}}_0$ based on constraint (\ref{P3_C1}). Finally, we present the overall progress of our second method in Algorithm \ref{ConvexProgramming}.

 \begin{algorithm}[h]
	\caption{Convex-Programming Method}\label{ConvexProgramming}
	\begin{algorithmic}[1]
	\State Solve $\mathscr{P}_2$ and $\mathscr{P}_3$ to obtain $\textbf{p}^*,\ \textbf{b}^*,\ \textbf{f}^* \text{and}\  \textbf{t}^*$
    \For{${j\in \mathcal{J}} $}\label{ForLoop}
    	\If{$f^*_{j}>f^{\text{max}}_j$}
        \State  \hspace{-0.5cm}	 $f^*_{j}\leftarrow f^{\text{max}}_j$
    	\State \hspace{-.5cm}	 Calculate the new task size $b^{+}_{j}$ using (\ref{NewTaskSize})
    	\State \hspace{-.5cm}	 $b^*_{j}\leftarrow b^{+}_{j}$
        \State \hspace{-.5cm}	 $b\leftarrow b-b^+_{j}$
        \State \hspace{-.5cm}	 $P^{\text{max}}\leftarrow P^{\text{max}}-P^{*}_{j
        }$
        \State \hspace{-.5cm}	 Disregard the $j$th device: $\mathcal{J}\leftarrow \mathcal{J}-\{j\} $
        \State  \hspace{-.5cm}	 Update $\textbf{p}^*,\ \hspace{-.1cm}\textbf{b}^*,\ \hspace{-.1cm}\textbf{f}^* \text{and}\  \textbf{t}^*$ \hspace{-.15cm}  by \hspace{-.15cm}  solving \hspace{-.15cm} \hspace{-.1cm}  $\mathscr{P}_2$\hspace{-.05cm} and \hspace{-.15cm} $\mathscr{P}_3$
        \State \hspace{-.5cm}	 Go to line \ref{ForLoop}
	    \EndIf\State \textbf{end}
	\EndFor \State \textbf{end}
    \State $\textbf{Output}\ \textbf{p}^*,\ \textbf{b}^*,\ \textbf{f}^* \text{and}\  \textbf{t}^*$
	\end{algorithmic}
\end{algorithm}
\noindent

\section{Simulation Results}\label{Sec:SimResults}
In this section, we compare the energy efficiency and run-time of the proposed methods through Monte Carlo simulations. In each simulation run, we uniformly place the offloading devices on a disk with a radius set to $15$m centered at the active device. Furthermore, we consider independent Rayleigh fading channels and distance-dependent path loss model, $\text{PL}\hspace{-0.05cm}=\hspace{-0.05cm}148 + 40\mbox{log}_{10}(d)$ in dB, where $d$ is the distance in km \cite{YDAI}. As a benchmark  we also include the energy consumption
when $\mathcal{J}\hspace{-0.05cm}=\hspace{-0.05cm}\emptyset$, i.e., the  task is completed locally. For the CPU throttling we assume that $\xi_i,\ i\in \mathcal{I},$ are uniform $\mathcal{U}(0,0.1)$, i.e., the actual computation resources may be below the allocated ones by up to $10\%$. We select the task size $b$ from a uniform distribution $\mathcal{U}(2\hspace{-0.06cm} \times\hspace{-0.06cm} 10^4\hspace{-0.05cm},\  \hspace{-0.1cm} 4\hspace{-0.06cm} \times\hspace{-0.06cm} 10^5 )$, and we set $f^{\text{max}}_0$ to a large value such that the assumption in the previous section holds. The rest of the system parameters are given in Table\hspace{-0.05cm} \ref{SysPar}.
\begin{table}[h]
	\vspace{0.17cm}
	\caption{System parameters} \label{SysPar}
	\resizebox{\columnwidth}{!}{%
		\begin{tabular}{ |l|c|c| }
			\hline
			\textbf{Parameter Description} & \textbf{Symbol} &\textbf{ Value} \\ \hline
			Number of offloading devices & $J$ & $ \{1, 2, 3\}	$ \\ \hline
			CPU cycles to process 1-bit data & $c$ & $ 1500\ \text{cycles}/\text{bit}	$ \\ \hline
	    	Effective  capacitance  constant & $ \kappa$ & $  10^{-24}\ $  W$\text{s}^3 $  \\ \hline
		    Max. iteration for DC prog. & $   k^{\text{max}}$ & $ 10^3	$ \\ \hline
			Max.  time  constraint & $t^{\text{max}}$ & $ [.4, 1]\text{s}	$ \\ \hline
			Max. transmit power & $	P^\text{max}$ & $  200\text{mW}	$ \\ \hline
			Limiting term for task uploading time & $ \alpha $ & $ .85$ \\ \hline
		    Reliability level  & $ \gamma $ & $ .95$ \\ \hline
		    Convergence tolerance for DC prog.  & $ \epsilon $ & $10^{-2} $ \\ \hline
		    Penalty parameter for DC prog.  & $ \lambda $ & $12$ \\ \hline
		    Max. radius of a D2D link  & - & 20m \\ \hline
		    Noise level & $N_0$ & $ -114	$dBm \\ \hline
		    Channel bandwidth & $W$ & $ 10	$MHz \\ \hline
		\end{tabular}	\label{SystemParameters}
	}
\end{table}
\begin{figure}[h]
	\centering
	\includegraphics[width=95mm]{Figs/Effect_TimeCons.eps}
	\small
	\caption{Expected total energy consumption versus $t^{\text{max}}$ for different numbers of offloading devices ($\gamma=.95$).}
	\normalsize
	\label{TotalEnergyvsMaxTime}
\end{figure}

In Fig. \ref{TotalEnergyvsMaxTime}, we investigate the effect of the maximum time limit to complete the task. We assume that the devices have the same $t^{\text{max}}$ and to simulate different computation capabilities of the offloading device we select $f^{\text{max}}_j$ from a uniform distribution $\mathcal{U}(3\times10^7, 1\times10^8)$ for each simulation run. As seen from Fig. \ref{TotalEnergyvsMaxTime}, regardless of the time limit, both of our methods significantly reduce the total energy consumption compared to local task computation. However, the performance of the convex-programming method outperforms DC programming in terms of energy efficiency. Specifically,  $t^{\text{max}}=0.4$s, the total energy consumption with our convex-programming method requires almost 30\% less energy to compute the same task with respect to computing it only at the local device. Note that by increasing the maximum time limit, we can reduce the required computation resources, which naturally lowers the energy consumption. However, this negatively impacts the quality of service of the given task in terms of latency.

\begin{figure}[h]
	\centering
	\includegraphics[width=95mm]{Figs/Effect_CompResource.eps}
	\small
	\caption{Expected total energy consumption versus $\text{F}_{\text{max}}$ for different numbers of offloading devices ($t^{\text{max}}=1$,  $\gamma=.95$).}
	\normalsize
	\label{TotalEnergyVSMax_f_ES}
\end{figure}

In Fig. \ref{TotalEnergyVSMax_f_ES}, we consider the effect of maximum computation resources at the offloading devices on the total energy consumption. Specifically, we select $f^{\text{max}}_j$ from a uniform distribution $\mathcal{U}(\text{F}_{\text{min}}, \text{F}_{\text{max}})$, where $\text{F}_{\text{min}}$ is set to $3\times10^7$ Hz while $\text{F}_{\text{max}}$ is ranging from $5\times10^7-1.5\times10^8$ Hz. Even though increasing the number of offloading devices drastically reduces the energy consumption, the amount of available computation resources at the offloading devices limits the energy efficiency during task offloading. Therefore, reducing the total energy consumption not only depends on the number of nearby devices but is also highly affected by the amount of available computation resources that can be allocated by the offloading devices.

Finally, in Table \ref{RunTimeComparison} we compare the average run-time of proposed methods implemented in \textsc{Matlab} on an Intel i7-3770 computer with 16GB RAM. The proposed convex-programming based method not only achieves a better performance compared to our DC programming approach in terms of energy efficiency but also its run-time is significantly shorter.
Specifically, with the increased number of offloading devices, DC programming takes at least ten times longer to converge within the selected tolerance value $\epsilon$. In addition, we observe that the required number of iterations for DC programming to converge is more than three times compared to our second method that is iteratively running Algorithm 2.
\begin{table}[h!]
	\caption{Average run-time comparison ($t^{\text{max}}=.4$)} \label{RunTimeComparison}
	\resizebox{\columnwidth}{!}{%
		\begin{tabular}{ |l|c|c|c| }
			\hline
        	\multicolumn{2}{|c|}{\textbf{Simulation setup} }  & \multicolumn{1}{c|}{\textbf{DC Prog. Method} }& \multicolumn{1}{c|}{\textbf{Convex-Prog. Method}}  \\ \hline
		    $\text{F}_{\text{max}}=4\times10^7 $ & $J=1$ & 3.98 \text{s}  & 0.40 \text{s} \\ \hline
		    $\text{F}_{\text{max}}=1\times10^8 $ & $J=1$ & 3.64 \text{s}  & 0.40  \text{s}  \\ \hline
		    $\text{F}_{\text{max}}=4\times10^7 $ & $J=2$ & 8.26 \text{s}  & 0.41  \text{s}  \\ \hline
		    $\text{F}_{\text{max}}=1\times10^8 $ & $J=2$ & 7.53 \text{s}  & 0.40  \text{s}  \\ \hline
		    $\text{F}_{\text{max}}=4\times10^7 $ & $J=3$ & 11.88 \text{s}  & 0.46 \text{s}  \\ \hline
		    $\text{F}_{\text{max}}=1\times10^8 $ & $J=3$ & 11.60 \text{s}  & 0.42 \text{s}  \\ \hline
		\end{tabular}
	}
\end{table}

\section{Conclusion}\label{Sec:Conclusion}
In this paper, we proposed two sub-optimal methods for a D2D-aided fog computing scenario under probabilistic time constraints. The first method relies on DC programming, however, its performance is very sensitive to the choice of the initial point. Hence, we propose a novel alternative solution based on convex programming, which eliminates the dependence on user-defined initialization. Nevertheless, due to the uncertainties on the allocated computation resources, we incorporate chance-constraint programming into both methods. While both proposed sub-optimal task offloading methods significantly reduce the total energy consumption compared to computing the task locally, the second method outperforms DC programming in terms of energy efficiency and run-time.

redpajama_set_name: RedPajamaArXiv