Abstract
In related research on queuing systems, in order to determine the system state, there is a widespread practice to assume that the system is stable and that distributions of the customer arrival ratio and service ratio are known information. In this study, the queuing system is looked at as a black box without any assumptions on the distribution of the arrival and service ratios and only keeping the assumption on the stability of the queuing system. By applying the principle of maximum entropy, the performance distribution of queuing systems is derived from some easily accessible indexes, such as the capacity of the system, the mean number of customers in the system, and the mean utilization of the servers. Some special cases are modeled and their performance distributions are derived. Using the chi-square goodness of fit test, the accuracy and generality for practical purposes of the principle of maximum entropy approach is demonstrated.
Highlights
Queuing theory is mainly regarded as a branch of applied probability theory
Let the mean number of customers in the system under steady state be Ls (§0), and pi is the probability of the fact that there are i(i~0,1,2,3, Á Á Á,) customers in the queuing system
According to the maximum entropy principle, the following model can be established if there is no more information
Summary
Queuing theory is mainly regarded as a branch of applied probability theory. Its applications are in different fields, such as communication networks, computer systems, machine plants, and services. Queuing theory tries to answer questions like the mean waiting time in the queue, the mean system response time (waiting time in the queue plus service time), the mean utilization of the service facility, the distribution of the number of customers in the queue, and the distribution of the number of customers in the system. These questions are mainly investigated in a stochastic scenario, where, for example, the inter-arrival times of the customers or the serving times are assumed to be random typically Poisson arrivals and to have exponent distribution serving times. Usually we are mainly interested in steady state solutions (see Figure 2); that is, where the system after a long running time tends to reach a stable state in which, for example, the distribution of customers in the system does not change (limiting distribution)
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