Computational analysis of a Markovian queueing system with geometric mean-reverting arrival process
Introduction
One approach to model a time-varying and volatile arrival process is to consider a Poisson process whose intensity is also a random process. This is known as a doubly stochastic Poisson process, or a Cox process (see, e.g. [7], [12], [2], [22]), in which the randomness in may bring in some desired features. The well-known MMPP (see, e.g. [8], [9], [13]) falls into this category in that the intensity process is modulated by a continuous-time Markov chain. With proper assumptions (e.g. phase-type distributed service time), a queueing system fed by an MMPP can be modeled by a large Markov chain for which some numerical techniques are readily available (e.g. the matrix geometric method of Neuts [18]). This makes MMPP a popular model for the traffic sources with random intensity.
Despite the convenience in modeling and the tractability in computation, there are some drawbacks of using MMPP as an arrival process. As commented in [10], there may be too many parameters involved in an MMPP. Consider the simplest case of a two-state MMPP, there are four parameters (transition rates between two states, and arrival rates at these states) to be fit (see, e.g. the fitting algorithm in [13]). But the number of parameters will grow dramatically (at an exponential rate) as the number of states increases and this limits the use of an MMPP with more states. In addition to the explosive growth in parameters, the lack of intuitive interpretations on these parameters (e.g. transition rates between states) also makes it difficult to understand their physical meanings. Furthermore, the abrupt change in its intensity makes the MMPP unsuitable for modeling the traffic sources with continuously changing random intensity or those without contrasting regimes (states).
From the perspective of a parsimonious model with interpretable parameters, the MMPP seems less competitive. Some new models are proposed to address these issues. One example is the discrete autoregressive (DAR) model (see, e.g. [10], [11], [15]) which is able to generate the geometric decaying autocorrelation with fewer parameters. The present study considers an alternative way to address the problems with MMPPs. We propose to use a geometric (GMR) mean-reverting diffusion process to model the intensity process. Such a process is continuous, positive, and mean-reverting, in that there is a long term mean and the process will be pulled back toward this mean when it deviates away. It is commonly used in finance for modeling asset prices with such features. For example, in [3], [19], [20], the GMR is used to model fuel and electricity prices. Here in the context of queueing, mean-reversion is motivated by the traffic control mechanism which will divert part of incoming traffic away when a line becomes heavily loaded, and will route some traffic back when it is less busy. Consider a queue in the middle stage of this line. When the traffic intensity goes up (down), the mechanism will bring an opposite effect against this upward (downward) trend. This causes the traffic intensity to be continuously changing and mean-reverting. Other motivations of using mean-reverting traffic sources can be found in recent studies. For example, [24] proposed to use a source of this kind to model “overdispersion” and “autocorrelation” as observed in call center traffic arrivals. Compared to modeling the intensity by a Markov chain, using GMR is able to reflect the smooth (instead of abrupt) changes in the intensity process. More importantly, the GMR is more parsimonious since fewer parameters are required to characterize the main features and each of them has clear physical interpretation. When the intensity is modulated by a GMR process, the arrival process is termed a GMR–MPP. The differences between Poisson processes with discrete and continuous intensities are depicted in Fig. 1.
To understand how such a continuous-state process influences the queueing performance, we consider the Markovian queueing system GMR–MPP/M/s. Because of the continuous nature in intensity, this system cannot be directly formulated as a large Markov chain. To make the existing numerical techniques applicable, we consider its counterpart system as an approximation, where the arrival process is replaced by DGMR(m)-MPP, a discrete-state (but still continuous-time) version of GMR-MPP. This system can be analyzed by the matrix-geometric method, but it remains to investigate how good the approximation is. Motivated by a theoretical convergence result, we establish a conjecture on the relation between the performance measures of the original and approximate systems. This conjecture is validated in the subsequent numerical analysis where the errors on the first fourth moments of queue length and waiting time are examined. Our results demonstrate that using the matrix geometric method together with Richardson extrapolation is able to provide very accurate and efficient estimates on the queueing performance. These numerical techniques are then applied to investigate the influences from the key traffic parameters on the queueing performance measures.
This rest of this paper is organized as follows. Section 2 introduces the GMR arrival process as well as its discrete-state counterpart. Section 3 formulates the counterpart queueing system as a large Markov chain and applies the matrix geometric method to analyze its performance. Section 4 presents the numerical results which validate our conjecture and investigate the influences from the GMR parameters. Finally Section 5 gives conclusions.
Section snippets
The geometric mean-reverting arrival process
This section introduces the geometric mean-reverting (GMR) process and defines the GMR–MPP arrival process. We also introduce its discrete counterpart in order to apply the existing numerical techniques. For the convenience of the subsequent queueing analysis, a conjecture is proposed and will be validated in our numerical study.
A process is said to follow a GMR diffusion process if it solves the following stochastic differential equation (see [19], [20])
Performance analysis of the GMR–MPP/M/s queueing system
To investigate how the GMR properties (e.g. speed of mean reversion, volatility) influence the queueing performance, we provide an analysis on the GMR–MPP/M/s system, where the service time of each customer is exponentially distributed with mean by one of the s servers. We assume that the system is stationary (i.e., it has already been at steady state) and the system load= for stability.
As suggested in the preceding section, we replace the GMR by its discrete counterpart
Numerical results
This section provides our numerical results which are divided into two parts. The first part examines the numerical efficiency and accuracy of the approximation. It includes a comparison of algorithms and the validation of our conjecture as presented in 4.1 A comparison of the algorithms for computing, 4.2 Validation of conjecture on convergence speed and the use of extrapolation The second part investigates how the queueing performance depends on the GMR–MPP parameters, particularly the
Conclusions
In this paper, we analyze the performance of a Markovian queueing system where the Poisson arrival is modulated by a continuous-state process known as GMR instead of a discrete-state process such as Markov chain. This GMR process is suitable for modeling the intensity processes which are smoothly changing with a mean-reverting nature. Compared with the commonly used MMPP, the proposed GMR–MPP is more parsimonious and its two key parameters σ and κ have clear physical meanings.
The proposed
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