Analysis of an SPP/G/1 system with multiple vacations and E-limited service discipline

Many researchers have studied variants of queueing systems with vacations. Most of them have dealt with M/G/1 systems and have explicitly analyzed some of their performance measures, such as queue length, waiting time, and so on. Recently, studies on queueing systems whose arrival processes are not Poissonian have appeared. We consider a single server queueing system with multiple vacations and E-limited service discipline, where messages arrive to the system according to a switched Poisson process. First, we consider the joint probability density functions of the queue length and the elapsed service time or the elapsed vacation time. We derive the equations for these pdf's, which include a finite number of unknown values. Using Rouché's theorem, we determine the values from boundary conditions. Finally, we derive the transform of the stationary queue length distribution explicitly.     

Bounds for the tail distribution in a queue with a superposition of general periodic Markov sources: theory and application

An efficient yet accurate estimation of the tail distribution of the queue length has been considered as one of the most important issues in call admission and congestion controls in ATM networks. The arrival process in ATM networks is essentially a superposition of sources which are typically bursty and periodic either due to their origin or their periodic slot occupation after traffic shaping. In this paper, we consider a discrete-time queue where the arrival process is a superposition of general periodic Markov sources. The general periodic Markov source is rather general since it is assumed only to be irreducible, stationary and periodic. Note also that the source model can represent multiple time-scale correlations in arrivals. For this queue, we obtain upper and lower bounds for the asymptotic tail distribution of the queue length by bounding the asymptotic decay constant. The formulas can be applied to a queue having a huge number of states describing the arrival process. To show this, we consider an MPEG-like source which is a special case of general periodic Markov sources. The MPEG-like source has three time-scale correlations: peak rate, frame length and a group of pictures. We then apply our bound formulas to a queue with a superposition of MPEG-like sources, and provide some numerical examples to show the numerical feasibility of our bounds. Note that the number of states in a Markov chain describing the superposed arrival process is more than 1.4 × 10⁸⁸. Even for such a queue, the numerical examples show that the order of the magnitude of the tail distribution can be readily obtained.     

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.     

A cyclic-service finite source model with round-robin scheduling

A cyclic-service finite source model with round-robin scheduling is considered. A single server scans sources in a cyclic manner whether there is a request or not. Each request consists of multiple tasks and only one task can be served when the server scans. We provide the exact analysis of the average response time in a heterogeneous finite source model, where the service times of tasks are distributed according to general probability distributions which may be different from one source to another. Further, we derive the Laplace-Stieltjes transform of the probability distribution function of the waiting time.     

Analysis of an Infinite-Server Queue with Batch Markovian Arrival Streams

This paper considers an infinite-server queue with multiple batch Markovian arrival streams. The service time distribution of customers may be different for different arrival streams, and simultaneous batch arrivals from more than one stream are allowed. For this queue, we first derive a system of ordinary differential equations for the time-dependent matrix joint generating function of the number of customers in the system. Next assuming phase-type service times, we derive explicit and numerically feasible formulas for the time-dependent and limiting joint binomial moments. Further, some numerical examples are provided to discuss the impact of system parameters on the performance.     

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.     

Loss probability in a finite discrete-time queue in terms of the steady state distribution of an infinite queue

We consider a discrete-time single-server queue with arrivals governed by a stationary Markov chain, where no arrivals are assumed to occur only when the Markov chain is in a particular state. This assumption implies that off-periods in the arrival process are i.i.d. and geometrically distributed. For this queue, we establish the exact relationship between queue length distributions in a finite-buffer queue and the corresponding infinite-buffer queue. With the result, the exact loss probability is obtained in terms of the queue length distribution in the corresponding infinite-buffer queue. Note that this result enables us to compute the loss probability very efficiently, since the queue length distribution in the infinite-buffer queue can be efficiently computed when off-periods are geometrically distributed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Distribution of spatial requirements for a MAP/G/1 queue when space and service times are dependent

In this paper, we consider a MAP/G/1 queue in which each customer arrives with a service and a space requirement, which could be dependent. However, the space and service requirements of different customers are assumed to be independent. Each customer occupies its space requirement in a buffer until it has completely received its service, at which time, it relinquishes the space it occupied. We study and solve the problem of finding the steady-state distribution of the total space requirement of all customers present in the system. In the process of doing so, we also generalize the solution of the MAP/G/1 queue and find the time-average joint distribution of the queue-length, the state of the arrival process and the elapsed service time, conditioned on the server being busy. This problem has applications to the design of buffer requirements for a computer or communication system.     

Multiclass Markovian fluid queues

This paper considers a multiclass Markovian fluid queue with a buffer of infinite capacity. Input rates of fluid flows in respective classes and the drain rate from the buffer are modulated by a continuous-time Markov chain with finite states. We derive the joint Laplace-Stieltjes transform for the stationary buffer contents in respective classes, assuming the FIFO service discipline. Further we develop a numerically feasible procedure to compute the joint and marginal moments of the stationary buffer contents in respective classes. Some numerical examples are then provided.