Mean field models are a popular tool used to analyse load balancing policies. In some exceptional cases the response time distribution of the mean field limit has an explicit form. In most cases it can be computed using either a recursion or a differential equation (for exponential job sizes with mean one). In this paper we study the value of the mean response time E[ R _\lambda] as the arrival rate \lambda approaches 1  (i.e. the system gets close to instability). As E[ R _\lambda] diverges to infinity, we scale with -\log(1-\lambda) and present a method to compute the limit \lambda\rightarrow 1^--E[ R _\lambda]/\log(1-\lambda). This limit has been previously determined for SQ(d) and LL(d), two well-known policies that assign an incoming job to a server with either the shortest queue or least work left among d randomly selected servers. However, the derivation of the result for SQ(d) relied on the closed form representation of the mean response time and does not seem to generalize well, moreover the proof for LL(d) is incomplete. In contrast, we present a general result that holds for any policy for which the associated recursion or differential equation satisfies a list of criteria. For SQ(d) and LL(d) these criteria are trivially verified. We apply our method to SQ(d,K) resp LL(d,K) with exponential job sizes of mean one. For these policies, jobs arrive in batches of size K and join the K servers with the shortest queue resp least amount of work left. For SQ(d ,K) we obtain \frac{1{\log(d/K)} as limiting value, while for LL(d,K) we find the limit to be }  equal to \frac{K}{d-K}. We further analyse a policy where SQ(d_i) resp LL(d_i) is used with probability p_i . For the shortest queue variant, we obtain the limit \frac{1{\log\left(\sum_{i=1}^np_id_i\right)}, while for the least loaded variant, we obtain \frac{1}{\sum_{i=1}^np_id_i-1}.} {\sum_ip_id_i-1}. We further indicate that our main result can also be used for load balancers with redundancy or memory. In addition, we propose an alternate scaling -\log(p_\lambda) instead of -\log(1-\lambda), for which the limit \lim_{\lambda\rightarrow 0^+}-E}[ -E}[ \log(1-\lambda) / \log(p_\lambda)=1.}
Mean field models are a popular tool used to analyse load balancing policies. In some cases the waiting time distribution of the mean field limit has an explicit form. In other cases it can be computed as the solution of a set of differential equations. Here we study the limit of the mean waiting time E[ W _\lambda] as the arrival rate \lambda approaches 1 for a number of load balancing policies when job sizes are exponential with mean 1 (i.e. the system gets close to instability). As E[ W _\lambda] diverges to infinity, we scale with -\log(1-\lambda) and present a method to compute the limit \lambda\rightarrow 1^--E[ W _\lambda]/\log(1-\lambda). This limit has a surprisingly simple form for the load balancing algorithms considered. We present a general result that holds for any policy for which the associated differential equation satisfies a list of assumptions. For the LL(d) policy which assigns an incoming job to a server with the least work left among d randomly selected servers these assumptions are trivially verified. For this policy we prove the limit is given by \frac{1 LL(d,K) {\log(d/K)} as limiting value, while for LL(d,K) we find the limit to be } policy, which assigns batches of K jobs to the K least loaded servers among d randomly selected servers, satisfies the assumptions and the limit is equal to \frac{K}{d-K}. For a policy which applies LL(d_i) with probability p_i {\log\left(\sum_{i=1}^np_id_i\right)}, while for the least loaded variant, we obtain \frac{1}{\sum_{i=1}^np_id_i-1}.} , we show that the limit is given by \frac{1{\sum_ip_id_i-1}. We further indicate that our main result can also be used for load balancers with redundancy or memory. In addition, we propose an alternate scaling -\log(p_\lambda) instead of -\log(1-\lambda), for which the limit \lim_{\lambda\rightarrow 0^+}-E}[W_\lambda /\log(p_\lambda) is well defined and non-zero (contrary to \lim_{\lambda\rightarrow 0^+-E}[W_\lambda /\log(1-\lambda)), while \lim_{\lambda\rightarrow 1^-\log(1-\lambda) / \log(p_\lambda)=1.}
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