Heavy Traffic Analysis of the Mean Response Time for Load Balancing Policies in the Mean Field Regime
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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_λ] as the arrival rate λ approaches 1 (i.e. the system gets close to instability). As E[R_λ] diverges to infinity, we scale with -log(1-λ) and present a method to compute the limit lim_λ→ 1^--E[R_λ]/log(1-λ). 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 1/log(d/K) as limiting value, while for LL(d,K) we find the limit to be equal to 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 1/log(∑_i=1^np_id_i), while for the least loaded variant, we obtain 1/∑_i=1^np_id_i-1.
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