Stationary queue length of a single-server queue with negative arrivals and nonexponential service time distributions

Abstract

Abstract. In this paper, a single-server queue with negative customers is considered. The arrival of a negative customer will remove one positive customer that is being served, if any is present. An alternative approach will be introduced to derive a set of equations which will be solved to obtain the stationary queue length distribution. We assume that the service time distribution tends to a constant asymptotic rate when time t goes to infinity. This assumption will allow for finding the stationary queue length of queueing systems with non-exponential service time distributions. Numerical examples for gamma distributed service time with fractional value of shape parameter will be presented in which the steady-state distribution of queue length with such service time distributions may not be easily computed by most of the existing analytical methods.

An alternative approach in finding the stationary queue length distribution of a queueing system with negative customers

“…The numerical method was first proposed in Koh [25] and has been successfully applied in some other queueing systems with different features. In Koh et al [23], the proposed method was applied in a model similar to this paper but the interarrival time of the positive customer remains exponentially distributed whereas the service time is relaxed to the one with CAR distribution. The work can be extended to queueing system with both the interarrival time and service time having a more general distribution.…”

“…We call such distribution a CAR distribution. Similar model has been considered in [23]. However, the service time for positive customer in [23] is assumed to have a non-exponential distribution while the interarrival time remains as exponentially distributed.…”

“…Similar model has been considered in [23]. However, the service time for positive customer in [23] is assumed to have a non-exponential distribution while the interarrival time remains as exponentially distributed. The same approach has been successfully applied to the models in [23][24][25][26][27].…”

“…However, the service time for positive customer in [23] is assumed to have a non-exponential distribution while the interarrival time remains as exponentially distributed. The same approach has been successfully applied to the models in [23][24][25][26][27]. The numerical result will be compared to the result computed using the analytical method presented in [24].…”

“…iI . By substituting(23) into(10),(11) and(13) to(16) when m = M -It can be seen that the process of substitution can be continue to Equation (12), (19) and (20) for m = M until all the Mjrk p are expressed in terms of 001 M p . The substitution process are then repeated for m = M -2, M -3, …, 1, 0 where of equations, we will get the value of 001 M p and substituting this value into (24), all stationary probabilities mjrk p will be obtained.…”