Stochastic Models

Borkovec, Milan; Dasgupta, Amites; Resnick, Sidney; Samorodnitsky, Gennady

doi: 10.1081/STM-120014217pmid: N/A

We model behavior of a TCP-like source transmitting over a single channel to a server that processes work at a constant rate τ. Transmission by the source follows an on/off mechanism. When the overall load in the system is below a critical constant γ, transmission rates increase linearly but when the load exceeds γ, then transmission rates decrease geometrically fast. We study the system by means of an embedded Markov chain, which gives the buffer content at the start of transmissions. Attention is paid to the time necessary to transmit a file of size L and both the tail behavior and expectation of the distribution of file transmission time are considered.

Chang, Seok Ho; Takine, Tetsuya; Chae, Kyung Chul; Lee, Ho Woo

doi: 10.1081/STM-120014218pmid: N/A

The main purpose of this article is to demonstrate a unified queue length formula for the BMAP/G/1 queue with generalized server vacations. We also present two unified formulas; one for the distribution of the workload, and the other for the joint distribution of the queue length and the remaining service time. Some factorizations, which are previously known for M/G/1 queues with vacations and for MAP/G/1 queue with multiple vacations, are now recast in this new framework. Further, based on these formulas, we provide general recursions for the performance measures of interest.

Sapna Isotupa, K. P.; Stanford, David A.

doi: 10.1081/STM-120014219pmid: N/A

This paper considers a single server queue that handles arrivals from N classes of customers on a non-preemptive priority basis. Each of the N classes of customers features arrivals from a Poisson process at rate λ i and class-dependent phase type service. To analyze the queue length and waiting time processes of this queue, we derive a matrix geometric solution for the stationary distribution of the underlying Markov chain. A defining characteristic of the paper is the fact that the number of distinct states represented within the sub-level is countably infinite, rather than finite as is usually assumed. Among the results we obtain in the two-priority case are tractable algorithms for the computation of both the joint distribution for the number of customers present and the marginal distribution of low-priority customers, and an explicit solution for the marginal distribution of the number of high-priority customers. This explicit solution can be expressed completely in terms of the arrival rates and parameters of the two service time distributions. These results are followed by algorithms for the stationary waiting time distributions for high- and low-priority customers. We then address the case of an arbitrary number of priority classes, which we solve by relating it to an equivalent three-priority queue. Numerical examples are also presented.

Emsermann, Markus; Simon, Burton

doi: 10.1081/STM-120014220pmid: N/A

In a simulation one can often identify a random variable, Y, that is likely to be highly correlated with a random variable of interest, X. If μ Y =E(Y) is known then Y can be used as a control variate to estimate μ X =E(X) more efficiently than by a direct simulation of X. We study the asymptotic properties of a method that uses Y to potentially speed up the simulation when μ Y is not known. The method is effective when μ Y can be efficiently estimated in an auxiliary simulation that does not involve X. We call Y a quasi control variate (QCV). For a simulation of length t>0 time units, we invest pt units estimating μ Y with the auxiliary simulation, yielding Y¯ pt . The remaining qt=(1−p)t units are spent on the main simulation yielding estimates (X˜ qt ,Y˜ qt ) for (μ X ,μ Y ). The two simulations can be interleaved so they are effectively done simultaneously. For each p∈(0,1) and α∈R we have a QCV estimator for μ X , We find p and α that minimize the asymptotic variance parameter (AVP) of Xˆ t (p,α) in terms of statistics that are estimated during the simulations, and then describe an easily implemented adaptive procedure that achieves the minimum AVP. The adaptive procedure evolves into the optimal QCV procedure if it is more efficient than a direct simulation, X¯ t →μ X ; otherwise it evolves into the direct simulation. Applications in stochastic linear programming, stochastic partial differential equations (PDE's) and queuing theory are cited.

Ye, Qiang

doi: 10.1081/STM-120014221pmid: N/A

We consider Latouche–Ramaswami's logarithmic reduction algorithm for solving quasi-birth-and-death models. We shall present some theoretical properties concerning convergence of the algorithm and discuss numerical issues arising in finite precision implementations. In particular, we shall present a numerically more stable implementation. A rounding error analysis together with numerical examples are given to demonstrate the higher accuracy achieved by the refined implementation.

Pflug, Georg; Rubinstein, Reuven Y.

doi: 10.1081/STM-120014222pmid: N/A

We consider a single-commodity, discrete-time, multiperiod (s, S)-policy inventory model with backlog. The cost function may contain holding, shortage, and fixed ordering costs. Holding and shortage costs may be nonlinear. We show that the resulting inventory process is quasi-regenerative, i.e., admits a cycle decomposition and indicates how to estimate the performance by Monte Carlo simulation. By using a conditioning method, the push-out technique, and the change-of-measure method, estimates of the whole response surface (i.e., the steady-state performance in dependence of the parameters s and S) and its derivatives can be found. Estimates for the optimal (s, S) policy can be calculated then by numerical optimization.