Abstract
To understand the performance of a queueing system, it can be useful to focus on the evolution of the content that is initially in service at some time. That necessarily will be the case in service systems that provide service during normal working hours each day, with the system shutting down at some time, except that all customers already in service at the termination time are allowed to complete their service. Also, for infinite-server queues, it is often fruitful to partition the content into the initial content and the new input, because these can be analyzed separately. With i.i.d service times having a non-exponential distribution, the state of the initial content can be described by specifying the elapsed service times of the remaining initial customers. That initial content process is then a Markov process. This paper establishes a many-server heavy-traffic (MSHT) functional central limit theorem (FCLT) for the initial content process in the space 𝔻𝔻, assuming a FCLT for the initial age process, with the number of customers initially in service growing in the limit. The proof applies a symmetrization lemma from the literature on empirical processes to address a technical challenge: For each time, including time 0, the conditional remaining service times, given the ages, are mutually independent but in general not identically distributed.
Highlights
Heavy-traffic (HT) functional central limit theorems (FCLT’s) for the standard G/G/s queueing model, with unlimited waiting space and service in order of arrival, expose the impact of the stochastic variability in the arrival and service processes on the transient and steady-state performance. This is important because the general G/G/s model is far less tractable than its Markovian M/M/s counterpart, even for the special case in which the interarrival times and service times come from independent sequences of i.i.d. random variables
From [14, 41], we know that conventional heavy-traffic theory tells a simple story: With conventional heavy-traffic, where the arrival rate increases to the maximum possible service rate with a fixed number of servers, the arrival and service processes contribute via Received February 2015
AMS 2000 subject classifications: Primary 60K25; secondary 60F17, 90B22, 37C55 Keywords and phrases: Heavy-traffic limits for queues, many-server queues, infiniteserver queues, terminating queues, time-varying arrivals, two-parameter processes, central limit theorem for non-identically distributed random variables
Summary
Heavy-traffic (HT) functional central limit theorems (FCLT’s) for the standard G/G/s queueing model, with unlimited waiting space and service in order of arrival, expose the impact of the stochastic variability in the arrival and service processes on the transient and steady-state performance. The results for the new arrivals come from [30], but unlike §5 of [30], Assumption 1 here makes the remaining service times at time 0 be conditionally independent, given the ages, but not identically distributed random variables. (t−y)+ 0 where Nis the limit process in the assumed FCLT for the arrival process specified in Assumption 1, and Kν(t, x) ≡ U (t, G(x)), with Ubeing a standard Kiefer process, capturing the variability of the new service times, and independent of N ; X1e,o is a zero-mean Gaussian process with the covariance function (3.7). (The joint convergence of Xn and Dn follows from continuous mapping theorem for addition at continuous limits.) In §4.1 we show that the main two-parameter process Xne,o can be decomposed into two other twoparameter processes Xne,,o1(t, y) and Xne,,o2(t, y), that can be treated separately by conditioning on the ages at time 0. In Step 3 (§4.2.3) we prove that Xne,,o1 is tight in DD
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