Exact sampling for some multi-dimensional queueing models with renewal input

Abstract

Using a recent result of Blanchet and Wallwater (2015: Exact sampling of stationary and time-reversed queues. ACM TOMACS, 25, 26) for exactly simulating the maximum of a negative drift random walk queue endowed with independent and identically distributed (iid) increments, we extend it to a multi-dimensional setting and then we give a new algorithm for simulating exactly the stationary distribution of a first-in-first-out (FIFO) multi-server queue in which the arrival process is a general renewal process and the service times are iid; the FIFO GI/GI/c queue with 2 ≤ c < ∞. Our method utilizes dominated coupling from the past (DCFP) as well as the Random Assignment (RA) discipline, and complements the earlier work in which Poisson arrivals were assumed, such as the recent work of Connor and Kendall (2015: Perfect simulation of M/G/c queues. Advances in Applied Probability, 47, 4). We also consider the models in continuous-time and show that with mild further assumptions, the exact simulation of those stationary distributions can also be achieved. We also give, using our FIFO algorithm, a new exact simulation algorithm for the stationary distribution of the infinite server case, the GI/GI/∞ model. Finally, we even show how to handle Fork-Join queues, in which each arriving customer brings c jobs, one for each server.Keywords exact sampling; multi-server queue; random walks; random assignment; dominated coupling from the past.

Sensitivity Analysis and Simulation of a Multiserver Queueing System with Mixed Service Time Distribution

“…First, the average working time of the algorithm may be infinite [36], e.g., in a system with large number of servers (which indeed depends on the regenerative cycle length). This problem can be solved either by the coupling-from-the-past technique [35] (which, although, is rather technically tricky), or by non traditional regenerative techniques, such as the artificial regeneration [41] or regenerative envelopes [40]. Finally, the study may be extended to larger classes of service time distributions.…”

“…A contribution of this work is that unlike classical discrete-event simulation (crude Monte-Carlo), which always has the so-called transient (warm-up) period during which an influence of initial conditions exists, we use the perfect simulation technique that allows exact sampling from the (unknown) steady-state distribution. In what follows, we rely on the regenerative approach designed for the M/G/c system in the work [15] (although there are recently developed more sophisticated techniques based on backward coupling, for instant [35], which are valid for a more general G/G/c system). Below, we outline the approach from [15].…”

“…We note that, although this approach allows to sample exactly from the steady-state distribution, the regeneration period in the dominated RA system can be very large in practice, and, thus, can lead to unacceptable long simulation. For further details on perfect sampling, see [15,[35][36][37]. Now we explain how to sample from the equilibrium distribution of a two-component mixture.…”

Omnithermal Perfect Simulation for Multi-server Queues

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A number of perfect simulation algorithms for multi-server First Come First Served queues have recently been developed. Those of Connor and Kendall (2015) and Blanchet, Pei, and Sigman (2015) use dominated Coupling from the Past (domCFTP) to sample from the equilibrium distribution of the Kiefer-Wolfowitz workload vector for stable M/G/c and GI/GI/c queues respectively, using random assignment queues as dominating processes. In this note we answer a question posed by Connor and Kendall (2015), by demonstrating how these algorithms may be modified in order to carry out domCFTP simultaneously for a range of values of c (the number of servers), at minimal extra cost.

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