We apply regenerative theory to derive certain relations between steady state probabilities of a Markov chain. These relations are then used to develop a numerical algorithm to find these probabilities. The algorithm is a modification of the Gauss-Jordan method, in which all elements used in numerical computations are nonnegative; as a consequence, the algorithm is numerically stable.

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We consider the economic behavior of a M/G/1 queuing system operating with the following cost structure: a server start-up cost, a server shut-down cost, a cost per unit time when the server is turned on, and a holding cost per unit time spent in the system for each customer. We prove that for the single server queue there is a stationary optimal policy of the form: Turn the server on when n customers are present, and turn it off when the system is empty. For the undiscounted, infinite horizon problem, an exact expression for the cost rate as a function of n and a closed form expression for the optimal value of n are derived. When future costs are discounted, we obtain an equation for the expected discounted cost as a function of n and the interest rate, and prove that for small interest rates the optimal discounted policy is approximately the optimal undiscounted policy. We conclude by establishing the recursion relation to find the optimal (nonstationary) policy for finite horizon problems.

The classification of stochastic processes { X i } into those with short or long-range dependence is based on the asymptotic properties of the variance of the sum S, = X I + X Z + . . + + X,. Suppose this process describes the number of packets (or cells) that arrive at a buffer; X t is the number that arrive in the tth time slice (e.g., 10 ms). We use a generic buffer model to show how the distribution of S, (for all values of m) determines the buffer occupancy. From this model we show that long-range dependence does not affect the buffer occupancy when the busy periods are not large. Numerical experiments show this property is present when data from four video conferences and two entertainment video sequences (which have long-range dependence) are used as the arrival process, even when the transmitting times are long enough to make the probability of buffer overflow 0.07. We generated sample paths from Markov chain models of the video traffic (these have shortrange dependence). Various operating characteristics, computed over a wide range of loadings, closely agree when the data trace and the Markov chain paths are used to drive the model. From this, we conclude that long-range dependence is not a crucial property in determining the buffer behavior of variable bit rate (VBR)-video sources.Index Terms-Asynchronous transfer mode (ATM), packet video, teleconferencing, broadband traffic.

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We consider situations where the server cannot continuously monitor its queue to sense customer arrivals. For this situation we introduce the T-policy which activates the server T time units after the end of the last busy period. We consider in detail an M/G/1 queue with linear customer holding costs and a fixed charge for activating the server. For the minimum cost-rate criterion we obtain the optimal value of T and the optimal cost rate. We show that the optimal cost rate is larger than the one achieved by the comparable optimal N-policy which activates the server when N customers are in the queue. We also show that under the optimal T-policy, the expected number of customers present when the server is activated is the optimal value of N.

In this paper we consider two-dimensional Markov chains with the property that, except for some boundary conditions, when the transition matrix is written in block form, the rows are identical except for a shift to the right. We provide a general theory for finding the equilibrium distribution for this class of chains. We illustrate theory by showing how our results unify the analysis of the M/G/1 and GI/M/1 paradigms introduced by M. F. Neuts.