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Approximating Response Time Distributions
Kimmo E. E. Raatikaineny
University of Helsinki, Dept. of Computer Science
Teollisuuskatu 23{25, SF-00510 Helsinki, Finland
Abstract: The response time is the most visible performance index to users of computer systems. End-users see individual response times, not the average. Therefore the distribution of response times is important in performance evaluation and capacity planning studies. However, the analytic results cannot be obtained in practical cases. A new method is proposed to approximate the response-time distribution. Unlike the previous methods the proposed one takes into account the service-time distributions and routing behaviour. The reported results indicate that the method provides reasonable approximations in many cases.
1 Introduction
Queueing network modelling is a popular tool in the performance evaluation of computer systems. It has been successfully used in various applications of modelling computer systems. In most applications only the mean values of performance indices, such as mean device queue-lengths and the mean system response-time, have been considered. The increasing usage of computer systems as production tools has raised the system response-time as the most important single performance indicator. People using the computer system see the response time as the only index of performance. The end-user experiences an individual response time, not the average. Therefore, the distribution of the system response-time, usually summarized by a set of quantiles or fractions, should receive greater interest.
Despite recent advances in analytic queueing network analysis, only for simple cases does an analytic technique give the distribution of the system response-time.
y This work was supported by Emil Aaltonen Foundation
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The most general cases are a cyclic queue and a network having a tree topology of exponential servers. The key assumption is the so-called overtaken-free condition [KePo83]. It means that any customer who arrives later than the tagged customer always stays behind him. The overtaken-free condition guarantees that the residence times at different nodes are independent. The residencetime distribution at a single node is available if the node is either a FCFS c-server or an IS server [McKe87].
There are two very simple queueing network models where the overtaken-free condition is not necessary. The first of them is a two-node network consisting of an IS center and a PS center. A detailed analysis is given in [Mitr81]. The second model is a three-node network of FCFS exponential servers, where there are two possible paths: 1 ! 2 ! 3 and 1 ! 3. This network is analyzed in [FaIM83].
Some approximate methods have also been proposed in [LaSe79], [SaLa81], and [ReSe88], for example. These methods can be used in slightly more complicated models. All the three methods are based on the same basic idea. The subnetwork the response-time distribution of which is analyzed is replaced by a load-dependent exponential server.
In this paper we introduce an alternative approach to approximate the response-time distribution in multiclass queueing network models having product-form solution. The method is based on the Laplace transform of an approximate response-time distribution. The response time is approximated as a random sum of residence times. The residence-time distribution of an IS- node is the same as its service-time distribution. The residence-time distribution of a PS- or FCFS-node is approximated through analyzing the node in isolation. The distribution of terms in the random sum is represented as the joint probability generating function.
The method includes three phases:
1. The Laplace transforms of the residence-time distribution for the PS- and FCFS-nodes are con-
190 Performance Evaluation Review Vol. 17 #1 May 1989