Multiple Arrival Streams Research Articles

Approximating Multiple Arrival Streams by Using Aggregation

In this paper we consider the superposition of independent Coxian arrival streams. We propose a method to efficiently approximate the superposition process, based on state space aggregation. The method is applied to a single server queue and to an inventory control model. Simulation results show that the method accurately estimates performance characteristics and that it significantly outperforms two-moment approximations.

Matrix Product-Form Solution for an LCFS-PR Single-Server Queue with Multiple Arrival Streams Governed by a Markov Chain

This paper considers a stationary single-server queue with multiple arrival streams governed by a Markov chain, where customers are served on an LCFS preemptive-resume basis. Service times of customers from each arrival stream are generally distributed and service time distributions for different arrival streams may be different. Under these assumptions, it is shown that the stationary joint distribution of queue strings representing from which arrival stream each customer in the system arrived and remaining service times of respective customers in the system has a matrix product-form solution, where matrices constituting the solution are given in terms of the infinitesimal generator of a certain Markov chain. Compared with the previous works, the result in this paper is more general in the sense that general service time distributions are allowed, and it has the advantage of computational efficiency. Note also that the result is a natural extension of the classical result for the LCFS-PR M/G/1 queue. Further, utilizing the matrix product-form solution, we derive a new expression of the vector Laplace–Stieltjes transform of the stationary distribution of unfinished work in the work-conserving single-server queue with multiple arrival streams governed by a Markov chain, which is given by the sum of matrix-geometric series.

Queue Length Distribution in a FIFO Single-Server Queue with Multiple Arrival Streams Having Different Service Time Distributions

This paper considers the queue length distribution in a class of FIFO single-server queues with (possibly correlated) multiple arrival streams, where the service time distribution of customers may be different for different streams. It is widely recognized that the queue length distribution in a FIFO queue with multiple non-Poissonian arrival streams having different service time distributions is very hard to analyze, since we have to keep track of the complete order of customers in the queue to describe the queue length dynamics. In this paper, we provide an alternative way to solve the problem for a class of such queues, where arrival streams are governed by a finite-state Markov chain. We characterize the joint probability generating function of the stationary queue length distribution, by considering the joint distribution of the number of customers arriving from each stream during the stationary attained waiting time. Further we provide recursion formulas to compute the stationary joint queue length distribution and the stationary distribution representing from which stream each customer in the queue arrived.

Distributional Form of Little's Law for FIFO Queues with Multiple Markovian Arrival Streams and Its Application to Queues with Vacations

This paper considers stationary queues with multiple arrival streams governed by an irreducible Markov chain. In a very general setting, we first show an invariance relationship between the time-average joint queue length distribution and the customer-average joint queue length distribution at departures. Based on this invariance relationship, we provide a distributional form of Little's law for FIFO queues with simple arrivals (i.e., the superposed arrival process has the orderliness property). Note that this law relates the time-average joint queue length distribution with the stationary sojourn time distributions of customers from respective arrival streams. As an application of the law, we consider two variants of FIFO queues with vacations, where the service time distribution of customers from each arrival stream is assumed to be general and service time distributions of customers may be different for different arrival streams. For each queue, the stationary waiting time distribution of customers from each arrival stream is first examined, and then applying the Little's law, we obtain an equation which the probability generating function of the joint queue length distribution satisfies. Further, based on this equation, we provide a way to construct a numerically feasible recursion to compute the joint queue length distribution.