Strong Approximations for Time-Dependent Queues
Mathematics of Operations Research
A time-dependent Mt/Mt/1 queue alternates through periods of under-, over-, and critical loading. We derive period-dependent, pathwise asymptotic expansions for its queue length, within the framework of strong approximations. Our main results include time-dependent fluid approximations, supported by a functional strong law of large numbers, and diffusion approximations, supported by a functional central limit theorem. This complements and extends previous work on asymptotic expansions of the
... xpansions of the queue-length transition probabilities. 1. Introduction. The governing laws for the evolution of real-world queueing systems vary with time. Yet queueing research and practice, spanning a period of over nine decades, have been devoted almost exclusively to time-homogeneous models. Such models can indeed provide reasonable approximations for slowly varying systems. However, there are many time-dependent phenomena, such as rush hour or periodicity, that they fail to capture. Time-dependent models are difficult to analyze, even in a Markovian setting. Our goal therefore, is to develop a rigorous framework for their asymptotic approximations, starting in this paper with the Mr/Mr/1 queue. A Markovian analysis of a time-homogeneous queueing system entails encoding its dynamics into Kolmogorov's forward (or backward) differential equations. Their solution yields the transition probabilities for the queueing model of the system. However, Kolmogorov's equations rarely have closed-form solutions, hence one resorts to steady state analysis. This reduces the problem from solving a set of differential equations to solving linear equations. The solution of the latter yields the steady state probabilities for the queueing model. Time-dependent queueing systems also can be modelled by continuous-time Markov chains, but they must be time-inhomogeneous. Their transition probabilities solve Kolmogorov's equations as well, but one cannot expect explicit solutions in view of the complexity already encountered in the time-homogeneous case. Worse still, it is not immediately clear what constitutes a steady state analysis for time-inhomogeneous systems (at least when its evolution is not periodic; see for example Asmussen and Thorisson (1987), Bambos and Walrand (1989), Harrison and Lemoine (1977), Heyman and Whitt (1984), Lemoine (1989), Rolski (1981, 1990)). In particular, approximating the behavior of the system in the here and now by its behavior at time "infinity" is typically futile. A time-inhomogeneous analogue to steady state analysis was proposed in the Ph.D. Thesis by Massey (1981) (see also Massey (1985) and Keller (1982)), where it was coined uniform acceleration. Here one scales all the average instantaneous transition rates of the Markovian model by a factor of 1/c. As e 0, each rate increases in *A. MANDELBAUM & W. A. MASSEY absolute terms, or is accelerated, but the ratio of any two rates relative to each other is held fixed. Uniform acceleration enables a dynamic asymptotic analysis of time-inhomogeneous queueing models, which yields asymptotic expansions that vary over time. Moreover, when applied to time-inhomogeneous Markovian systems, it reduces to either steady state or heavy traffic analysis. (See the examples in ??4.1 and 4.4.) In Massey (1985), uniform acceleration gave rise to an asymptotic expansion of the transition probabilities for the queue length process of a time dependent M/M/1 queue, hereafter denoted by Mt/Mt/1. This provided a rigorous foundation to the earlier work of Newell (1968) and Keller (1982). It also led to the proper notion of a time-dependent traffic intensity parameter, namely p*(t) defined in (3.1) below and elaborated on in ?7. The purpose of this paper is to complement and refine Newell (1968), Massey (1981, 1985) and Keller (1982). We do this through an asymptotic analysis of the queue length sample paths, within the unifying framework of the strong approximation theorems, introduced by Koml6s, Major and Tusnady (1976). Strassen was the first to prove a strong approximation result, then Skorohod introduced his embedding of random walks in a Brownian motion, and Keifer used it to establish answers to questions about best convergence rates (see Cs6rgo and Revesz (1981) and Cs6rgo and Horvath (1993) for a survey on the historical evolution of the subject). However, the framework operate within is the one by Koml6s, Major, and Tusnady (1976). Specifically, in ?2, we apply uniform acceleration directly to the sample paths (2.1) of the queue Mt/Mt/1. The outcome is the asymptotic expansion (2.7). Its derivation relies on a functional strong law of large numbers (FSLLN, Theorem 2.1) and a functional central limit theorem (FCLT, Theorem 2.2). Both theorems are consequences of the strong approximation results in Theorem 2.3. The FSLLN limit (2.4) is deterministic and, as shown later, has the interpretation of a fluid flow system. Viewing the original model as a microscopic description, this deterministic fluid model provides a macroscopic fluid approximation of the queue which, furthermore, is the zeroth-order term in the asymptotic expansion (2.7) of its sample paths. The stochastic FCLT limit (2.2) then deserves to be referred to as a mesoscopic first-order refinement of the fluid model. During its evolution, the Mt/Mt/1 queue can alternate between underloaded, critically loaded and overloaded phases. These phases are determined by its fluid approximation, and the phase transitions are summarized in Figure 3.1. Moreover, the asymptotic expansion (2.7) can be localized to each phase, and this is outlined in ?3 and substantiated in ??8-10. In ?4, we specialize our results to the time-homogeneous M/M/1 queue and to two periodic models. Finally, ??5 and 6 are devoted to proving Theorems 2.1-2.3 and their supporting assertions. Of special importance is Lemma 5.2, which provides the sample-path intuition behind our main asymptotic expansion (2.7). The literature on time-inhomogeneous models, like the MM/Mt/1 queue, is not vast. Insight and calculations have been commonly based on either approximations (Luchak (1956), Newell (1968), Keller (1982), Massey (1981, 1985) , Rothkopf and Oren (1979), for example), or simulation (Green, Kolesar, and Svornos (1991) for example). Exact results are rarely available, with the notable exception of networks with Poisson arrivals and infinite server nodes (see Eick, Massey and Whitt (1993a, b) as well as Massey and Whitt (1993)). For a textbook treatment of some aspects of time-dependent queues, see Hall (1991), for example. Our paper focuses only on Poissonian single-stations, but we are also studying time-inhomogeneous Markovian networks (Mandelbaum and Massey, in preparation), for which the current paper is a prerequisite. Our framework also accommodates more general point processes, as in Chap- STRONG APPROXIMATIONS FOR TIME-DEPENDENT QUEUES ter 10 of Lipster and Shiryaev (1989). This latter work employs uniform acceleration of state-dependent queues, as also in Anulova (1989), Krichagina, Lipster, and Puhalski (1988), and Yamada (1984). Finally we note that the first application of the Komlos, Major and Tusnady theorem is due to Rosenkrantz (1980). The results of paper suggest that it might be possible to obtain estimates, in terms of c, on the rates of convergence of the distributions of certain functionals to their limits.