On the Shortest Queue Version of the Erlang Loss Model

Abstract

We consider two parallel M/M/N/N queues. Thus there are N servers in each queue and no waiting line(s). The network is fed by a single Poisson arrival stream of rate λ, and the 2N servers are identical exponential servers working at rate µ. A new arrival is routed to the queue with the smaller number of occupied servers. If both have the same occupancy then the arrival is routed randomly, with the probability of joining either queue being 1/2. This model may be viewed as the shortest queue version of the classic Erlang loss model. If all 2N servers are occupied further arrivals are turned away and lost. We let ρ = λ/µ and a = N /ρ = N µ/λ. We study this model both numerically and asymptotically. For the latter we consider heavily loaded systems (ρ → ∞) with a comparably large number of servers (N → ∞ with a = O(1)). We obtain asymptotic approximations to the joint steady state distribution of finding m servers occupied in the first queue and n in the second. We also consider the marginal distribution of the number of occupied servers in the second queue, as well as some conditional distributions. We show that aspects of the solution are much different according as a > 1/2, a ≈ 1/2, 1/4 < a < 1/2, a ≈ 1/4 or 0 < a < 1/4. The asymptotic approximations are shown to be quite accurate numerically.

The shorter queue polling model

“…There exists an extensive literature on dispatching policies and their optimality [9,26,27,35,36,37,42,46]. Among the dispatching policies, the join-the-shortest-queue (JSQ) policy has received considerable attention [5,6,10,19,20,23,24,28,30,44,45]. The JSQ policy in some scenarios has been proven to be the optimal policy; on the one hand it minimizes the customers mean waiting time [25] and on the other hand it stochastically maximizes the number of customers served by time t, t > 0 [43].…”

On the Nonsymmetric Longer Queue Model: Joint Distribution, Asymptotic Properties, and Heavy Traffic Limits

“…and we also note that the expressions in Theorem 1 are consistent with (22). We will show that the asymptotics are quite different whether…”

“…In addition to this model being interesting on its own, many variants of shortest queue problems, such as ones with multiple servers and finite capacities, can be asymptotically reduced to LQ models of the type considered here (see [22,23]). For example, in [23] we showed that the finite capacity version of the standard symmetric SQ model (analyzed in [4,5]), where 𝑁 1 , 𝑁 2 ⩽ 𝐾 and 𝐾 is the capacity, asymptotically reduces to the symmetric LQ model in [1], if we consider the process (𝐾 − 𝑁 1 , 𝐾 − 𝑁 2 ), which measures the number of spots available in the two waiting rooms.…”

Optimisation of Server Selection for Maximising Utility in Erlang-Loss Systems

“…A different approach to the M/M/n/n can be found in [8], where Yao & Knessl outlined two parallel M/M/n/n queues with n servers in each queue and no waiting lines. If both have the same occupancy, then the arrival is routed to any of them.…”