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

A single-server queueing system with negative customers is considered in this paper. One positive customer will be removed from the head of the queue if any negative customer is present. The distribution of the interarrival time for the positive customer is assumed to have a rate that tends to a constant as time t tends to infinity. An alternative approach will be proposed to derive a set of equations to find the stationary probabilities. The stationary probabilities will then be used to find the stationary queue length distribution. Numerical examples will be presented and compared to the results found using the analytical method and simulation procedure. The advantage of using the proposed alternative approach will be discussed in this paper.

Analysis of an N–Policy GI/M/1 Queue in a Multi–Phase Service Environmentwith Disasters

This paper investigates an N-policy GI/M/1 queue in a multi-phase service environment with disasters, where the system tends to suffer from disastrous failures while it is in operative service environments, making all present customers leave the system simultaneously and the server stop working completely. As soon as the number of customers in the queue reaches a threshold value, the server resumes its service and moves to the appropriate operative service environment immediately with some probability. We derive the stationary queue length distribution, which is then used for the computation of the Laplace-Stieltjes transform of the sojourn time of an arbitrary customer and the server’s working time in a cycle. In addition, some numerical examples are provided to illustrate the impact of several model parameters on the performance measures.

Repairable Queue with Non-exponential Interarrival Time and Variable Breakdown Rates

This paper considers a single server queue in which the service time is exponentially distributed and the service station may breakdown according to a Poisson process with the rates γ and γ' in busy period and idle period respectively. Repair will be performed immediately following a breakdown. The repair time is assumed to have an exponential distribution. Let g(t) and G(t) be the probability density function and the cumulative distribution function of the interarrival time respectively. When t tends to infinity, the rate of g(t)/[1 – G(t)] will tend to a constant. A set of equations will be derived for the probabilities of the queue length and the states of the arrival, repair and service processes when the queue is in a stationary state. By solving these equations, numerical results for the stationary queue length distribution can be obtained.

On the Discrete-TimeGeo/G/1Queue with Vacations in Random Environment

A discrete-timeGeo/G/1queue with vacations in random environment is analyzed. Using the method of supplementary variable, we give the probability generating function (PGF) of the stationary queue length distribution at arbitrary epoch. The PGF of the stationary sojourn time distribution is also derived. And we present the various performance measures such as mean number of customers in the system, mean length of the type-icycle, and mean time that the system resides in phase0. In addition, we show that theM/G/1queue with vacations in random environment can be approximated by its discrete-time counterpart. Finally, we present some special cases of the model and numerical examples.

Analysis of General Input State Dependent Working Vacation Queue with Changeover Time

We consider a finite buffer GI/M(n)/1 queue with multiple working vacations and changeover time, where the server can keep on working but at a slower speed during the vacation period. Moreover, the amount of service demanded by a customer is conditioned by the queue length at the moment service is begun for that customer. We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. Finally, some numerical results of the model are presented to show the parameter effect on various performance measures.

EXPLICIT SOLUTION FOR QUEUE LENGTH DISTRIBUTION OF M/T-SPH/1 QUEUE

In this paper we study an M/T-SPH/1 queue system, where T-SPH denotes the continuous time phase type distribution defined on a birth and death process with countably many states. The queue model can be described by a quasi-birth-and-death (QBD) process with countable phases. For the QBD process, we give the computation scheme of the joint stationary distribution. Furthermore, the obtained results enable us to give the stationary queue length distribution for the M/T-SPH/1 queue.

Optimal control for a BMAP/G/1 queue with two service modes

Queueing models with controllable service rate play an important role in telecommunication systems. This paper deals with a single-server model with a batch Markovian arrival process (BMAP) and two service modes, where switch-over times are involved when changing the service mode. The embedded stationary queue length distribution and the explicit dependence of operation criteria on switch-over levels and derived.

Optimal Control for an Mx/G/1 Queue with Two Operation Modes

The controlled Mx/G/1-type queueing model with two modes of operation is considered. The modes are characterized by different service time distributions and input rates. The switchover times are imposed in the model. The embedded stationary queue-length distribution and the explicit dependence of operation criteria on switchover levels are derived.