We develop a model to quantify the performance of message transmission systems in which users must reserve transmission resources via a contention mechanism prior to transmission. Our work is motivated by a desire to understand the performance characteristics of systems such as the general packet radio service (GPRS), where the single forward link of the wireless access system is organized as a sequence of frames, each of which has first a contention period and then a service period. There are a fixed number of fixed-length contention slots in each contention period. Each contending customer chooses at random the slot in which to contend, and success is determined by a capture model. A contender who fails waits for the next contention period, then again chooses at random the slot in which to contend; this process is repeated until the contender is successful. Customers who have contended successfully are served during the service period, which has a prescribed number of fixed-length slots, on a first-come-first-served (FCFS) basis, with the required number of service units being drawn independently from a general discrete distribution having finite support. We model the system as a Markov renewal process embedded at service departure times. We solve the model and then compute the equilibrium distributions of the number of customers in the system at arbitrary points in time and at customer arrival times. Finally, we give a numerical example in which we demonstrate the usefulness of our results in understanding the behavior of GPRS.

In this paper, we consider a single-server service system with contention periods preceding the service operation. Customer interarrival times are exponentially distributed with a rate that depends on the current system occupancy, and customers must be successful in the contention phase in order to obtain access to the service phase. There are C contention slots in each contention period, which are chosen at random by the contending customers, and success is determined by a capture model. Each contention period is followed by S service periods. Customers who have already contended successfully receive service on a FCFS basis, and service times are drawn independently from a discrete distribution. We model the system as a Markov renewal process embedded at departure service times. We solve the model and then compute the equilibrium distributions of the number of customers in the system at an arbitrary point in time and at customer arrival epochs. Finally, we give numerical examples that show the effects of contention on system performance.

The queueing behavior of many communication systems is well modeled by a queueing system in which time is slotted, and the number of entities that arrive during a slot is dependent upon the state of a discrete time, discrete state Markov chain. Techniques for analyzing such systems have appeared in the literature from time to time, but distributions have been presented in only rare instances In the paper, the authors present the probability generating function (PGF) for joint and marginal buffer occupancy distributions of statistical time division multiplexing systems in this class. They discuss inversion of the PGF using discrete Fourier transforms, and also discuss a simple technique for obtaining moments of the queue length distribution. Numerical results, including queue length distributions for some special cases, are presented.>

The queuing behavior of many communication systems is well modeled by a queuing system in which time is slotted and in which the number of entities arriving during a slot is dependent upon the state of a discrete-time, discrete-state Markov chain. The probability generating function is presented for joint and marginal buffer occupancy distributions of statistical time-division multiplexing systems in this class. A simple technique is discussed for obtaining moments of the queue length distribution. In addition, a discussion is presented of inversion of the probability generating function. Numerical results, including queue length distributions for some special cases, are presented.>