Higher-order distributional properties in closed queueing networks

Abstract

In many real-life computer and networking applications, the distributions of service times, or times between arrivals of requests, or both, can deviate significantly from the memoryless negative exponential distribution that underpins the product-form solution for queueing networks. Frequently, the coefficient of variation of the distributions encountered is well in excess of one, which would be its value for the exponential. For closed queueing networks with non-exponential servers there is no known general exact solution, and most, if not all, approximation methods attempt to account for the general service time distributions through their first two moments.

We consider two simple closed queueing networks which we solve exactly using semi-numerical methods. These networks depart from the structure leading to a product-form solution only to the extent that the service time at a single node is non-exponential. We show that not only the coefficients of variation but also higher-order distributional properties can have an important effect on such customary steady-state performance measures as the mean number of customers at a resource or the resource utilization level in a closed network.

Additionally, we examine the state that a request finds upon its arrival at a server, which is directly tied to the resulting quality of service. Although the well-known Arrival Theorem holds exactly only for product-form networks of queues, some approximation methods assume that it can be applied to a reasonable degree also in other closed queueing networks. We investigate the validity of this assumption in the two closed queueing models considered. Our results show that, even in the case when there is a single non-exponential server in the network, the state found upon arrival may be highly sensitive to higher-order properties of the service time distribution, beyond its mean and coefficient of variation.

This dependence of mean numbers of customers at a server on higher-order distributional properties is in stark contrast with the situation in the familiar open $M/G/1$ queue. Thus, our results put into question virtually all traditional approximate solutions, which concentrate on the first two moments of service time distributions.

Introduction

The objective of this paper is to show that there can be big discrepancies between exact results and traditional approximations due to the influence of distributional properties of inter-arrival and service times on the performance of queueing networks. Here, we consider two very simple closed queueing networks which deviate from the product form only in that a single node is non-exponential. We examine customary steady-state performance metrics (mean number of requests at a server, server utilization), as well as the degree of departure from the Arrival Theorem.

Since in many real-life situations the service and/or inter-arrival times tend to exhibit high variability (e.g. due to the use of caching, or intrinsic nature of certain types of Internet traffic [1]), we focus on the case where the coefficient of variation of the service time exceeds one. Using a recently-developed semi-numerical solution method [2] and its generalization [3], we show that the state found upon arrival and customary steady-state performance metrics may exhibit important dependence on higher-order properties of the service time distribution. Such dependence casts a doubt over the value of approximations traditionally limited to the first two moments of the distribution.

In a large number of real-life computer and networking applications, the state of a resource that a request finds upon its arrival at the resource greatly impacts the resulting quality of service. To some extent, what an arriving request “sees” may be viewed as more important than the customary steady-state performance metrics such as the mean number of requests or server utilization. As an example, from the standpoint of an I/O request generated by the host the probability that the requests find a free I/O path is a more critical performance measure than the overall path utilization.

The Arrival Theorem [4], [5], [6] for closed product-from networks states that the state found upon arrival is the same as the steady state of the network without the arriving request. This theorem is at the heart of the Mean Value Analysis of queueing networks [7], [8], [9], [10]. The elegant simplicity of the Arrival Theorem makes it an attractive basis for approximations even when the network does not posses a product-form solution [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].

To the best of our knowledge, there is a limited number of studies attempting to quantify the degree of applicability of the Arrival Theorem in networks with non-exponential service times [22], [23], [19], [24]. Possibly inspired by the distributional dependence factor in the Pollaczek–Khintchine formula [25], [26] for the $M/G/1$ queue, most existing studies seem to concentrate on the influence of the coefficient of variation of the service time distribution [22], [23], [19]. This appears to be the case as well in some attempts to improve the approximation given by the “raw” Arrival Theorem by introducing corrective terms related to the first two moments of the service time distribution [12], [13], [20], [21]. The influence of properties of order higher than two (such as skewness and kurtosis) of the service time distribution seems to have attracted little attention [27], [28], [22].

Our contribution is threefold. First, we show that, even for a very simple closed network with just a single non-exponential server, the performance of the system may depend in an important way on higher-order properties, beyond the first two moments, of the service time distribution. This provides evidence that many traditional approximations for non-exponential closed queueing networks (e.g.  [29], [30], [31], [32], [33], [20]) need to be re-evaluated. Second, we examine the degree of applicability of the Arrival Theorem as a function of both the distribution of the service times and the number of users in the system. Our results provide some indication when the theorem can be expected to be a reasonable approximation, and when the deviation from it can be almost arbitrarily large. Third, we show that the influence of higher-order properties is not limited to the skewness of the service time distribution but includes properties of even higher order.

This paper is organized as follows. In Section 2 we describe the first queueing network considered and we present our numerical results for this system. Section 3 is devoted to the second simple network and its numerical results. Section 4 concludes this paper.