369 research outputs found
Single server tandem queues with finite buffers: An improved decomposition method
No full textThis paper analyzes a single-server tandem queue with generally distributed service times, finite buffers and blocking. We will construct an iterative method to approximate the queue length distribution of each buffer, by refining the ideas of Van Vuuren and Adan (2009), where the tandem queue is decomposed into single-buffer subsystems. In each subsystem we incorporate possible starvation (of the upstream buffer) in the service time of the first server and possible blocking (of the downstream buffer) in the service time of the second server. The re??nement is the two-dimensional Markov description of the service time of the second server, where one dimension keeps track of the elapsed service time and the other one keeps track of the elapsed blocking time (if any). The starvation and blocking times are determined iteratively. Numerical results show an improvement of the performance estimates (such as throughput and mean sojourn times) compared to the method of Van Vuuren and Adan (2009), where significant improvements are observed in the zero-buffer cases
A loss system with skill based servers under assign to longest idle server policy
Full text linkWe consider a memoryless loss system with servers S = {1,…, J}, and with customer types C = {1,…,I}. Servers are multi-type: server j works at rate µ_j, and can serve a subset of customer types C(j). An arriving customer will go to the longest idling server which can serve him, or be lost. We obtain a simple explicit steady state distribution for this system, and calculate various performance measures of this system in steady state. We provide some illustrative examples. We compare this system with a similar system discussed recently by Adan, Hurkens and Weiss [2]. We also show that this system is insensitive, the results hold also for general processing time distributions.
Keywords: Service system; loss system; multi type customers; multi type servers; product form solution; go to longest waiting server policy; insensitivity
Analysing multiprogramming queues by generating functions
Full text linkThe generating function approach for analysing queueing systems has a longstanding tradition. One of the highlights is the seminal paper by Kingman on the shortest queue problem, where the author shows that the equilibrium probabilities P_{m,n} of the queue lengths can be written as an infinite sum of products of powers. The same approach is used by Hofri to prove that for a multiprogramming model with two queues the boundary probability P_{0,n} can be expressed as an infinite sum of powers. The present paper shows that the latter representation does not always hold, which implies that the multiprogramming problem is essentially more complicated than the shortest queue problem. However, it appears that the generating function approach is very well suited to show when such a representation is available and when not
Analyzing GI/E_r/1 queues
No full textIn this paper we study a single-server system with Erlang-r distributed service times and arbitrarily distributed interarrival times. It is shown that the waiting-time distribution can be expressed as a finite sum of exponentials, the exponents of which are the roots of an equation. Under certain conditions for the interarrival-time distribution, this equation can be transformed to r contraction equations, the roots of which can easily be found by successive substitutions. The conditions are satisfied for several practically relevant arrival processes. The resulting numerical procedures are easy to implement and efficient and appear to be remarkably stable, even for extremely high values of r andfor values of the traffic load close to 1. Numerical results are presented
A class of Markov processes on a semi-infinite strip
Full text linkIn this paper we determine the equilibrium distribution for a general class of Markov processes on a semi-infinite strip. We expose a method to express the equilibrium probabilities of the Markov process as a finite sum of terms, which are geometric in the unbounded variable. The geometric factors are the roots inside the unit circle of a determinantal equation. By using a generating-function technique we are able to determine these roots very efficiently. Because of this, the expression for the equilibrium probabilities becomes numerically very attractive
How heavy-tailed distributions affect simulation-generated time averages
No full textFor statistical inference based on telecommunications network simulation, we examine the effect of a heavy-tailed file-size distribution whose corresponding density follows an inverse power law with exponent a + 1, where the shape parameter a is strictly between 1 and 2. Representing the session-initiation and file-transmission processes as an infinite-server queueing system with Poisson arrivals, we derive the transient conditional mean and covariance function that describes the number of active sessions as well as the steady-state counterparts of these moments. Assuming the file size (service time) for each session follows the Lomax distribution, we show that the variance of the sample mean for the time-averaged number of active sessions tends to zero as the power of 1 - a of the simulation run length. Therefore, impractically large sample-path lengths are required to achieve point estimators with acceptable levels of statistical accuracy. This study compares the accuracy of point estimators based on the Lomax distribution with those for lognormal and Weibull file-size distributions whose parameters are determined by matching their means and a selected extreme quantile with those of the Lomax. Both alternatives require shorter run lengths than the Lomax to achieve a given level of accuracy. Although the lognormal requires longer sample paths than the Weibull, it better approximates the Lomax and leads to practicable run lengths in almost all scenarios
Analysis of a simple Markovian re-entrant line with infinite supply of work under the LBFS policy
No full textWe consider a two machine 3 step re-entrant line, with an infinite supply of work. The service discipline is last buffer first served. Processing times are independent exponentially distributed. We analyze this system, obtaining steady state behavior and sample path properties
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