Asymptotically Optimal Routing and Servive Rate Allocation in a Multiserver Queueing System

Abstract

We consider a single stage queueing system with c heterogeneous servers. Customers arrive at this system according to a renewal process with mean 1/λ and squared coefficient of variation (scv) Ca2. An incoming customer is routed to server i with probability θi, ∑i=1cθi = 1. The service times at server i are i.i.d random variables with mean 1/μi, and scv Csi2. The holding cost rate of queue i is hi per customer, i = 1, 2, …, c. The problems of interest are twofold: (a) for a fixed service rate allocation μi, ∑i=1c μi = μ, find the routing probabilities, θi*, ∑i=1c θi* = 1, that minimize the average total holding cost; and (b) for fixed routing probabilities θi, ∑i=1c θi, and total service rate μ, find the service rate allocation μi* = μδi*, ∑i=1c δi* = 1, that minimizes the average total holding cost of the system. For each problem, we characterize the optimal policy under heavy traffic conditions. We also derive the routing probabilities, θ̂i (proportions δ̂i), i = 1, …, c, that are strongly asymptotically optimal. That is, the difference between the average total holding costs under θ̂i, i = 1, …, c, and θi*, i = 1, …, c (δ̂i, i = 1, …, c, and δi*, i = 1, …, c) is bounded by a fixed constant independent of the routing probabilities (proportions) and the arrival rate. In addition, we discuss the necessity and sufficiency of the accurate knowledge of the means and scvs of the interarrival and service times m obtaining asymptotically optimal policies.

Control of parallel non-observable queues: Asymptotic equivalence and optimality of periodic policies

“…: p ∈ S, for S compact and convex, and we stress that EW (p) is a convex function of p. Using a classic result in convex optimization, this means that a polynomial number of evaluations of the objective function EW (p) are sufficient to converge to an optimizer of the problem. Given that each objective evaluation is efficient [30,34,32,13,14,8], this lets us conclude that we have significantly reduced much of the difficulty of Problem 1. In the case where service times have an exponential distribution, EW (p) admits a very simple characterization because it is the weighted mean waiting time of R D/M/1 queues [9,4].…”

Optimal allocation of resources in a job shop environment

“…The form of the expression above is no coincidence as demonstrated by Shanthikumar and Xu [29] who considered a single stage queueing system with heterogeneous servers. Customers arrive according to a renewal process.…”

Optimal service‐capacity allocation in a loss system

“…See Shanthikumar and Xu [38], and §6 in Altman [5]. Korilis, Lazar, and Orda [30] (see also [29]) consider a finite number of users, each wishing to minimize the expected delay by splitting demand among a set of parallel heterogeneous M/M/1 servers with known service rates.…”