Bounds For Performance Characteristics; A Systematic Approach Via Cost Structures

Communications in Statistics. Part C: Stochastic Models 02/1999; 14(1-2). DOI: 10.1080/15326349808807467
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In this paper we present a systematic approach for the construction of bounds for the average cost in Markov chains with infinitely many states. The technique to prove the bounds is based on dynamic programming. Most performance characteristics of Markovian systems can be represented by the average cost for some appropriately chosen cost structure. Therefore, the approach can be used to generate bounds for relevant performance characteristics. The approach is demonstrated for the shortest queue model. It is shown how for this model several bounds for the mean waiting time can be constructed. We include numerical results to demonstrate the quality of these bounds. 1 INTRODUCTION In this paper we consider an irreducible N-dimensional Markov chain with states m = (m 1 ; Delta Delta Delta ; mN ), where each m i is an integer, and transition probabilities p(m;n). Let ß denote its equilibrium distribution (which we assume to exist). If c(m) is the cost per period in state m, then the ...

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Available from: Ivo Adan, May 30, 2013
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    • "Earlier methods applicable for ordering of flows are mostly based on Markov reward comparison techniques (e.g. [1] [3] [12] [13] [14]). While more limited in scope than the general Markov reward approach, the flow coupling technique presented here, when applicable, yields stronger ordering results using simpler analysis. "
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    ABSTRACT: Robust estimates for the performance of complicated queueing networks can be obtained by showing that the number of jobs in the network is stochastically comparable to a simpler, analytically tractable reference network. Classical coupling results on stochastic ordering of network populations require strong monotonicity assumptions which are often violated in practice. However, in most real-world applications we care more about what goes through a network than what sits inside it. This paper describes a new approach for ordering flows instead of populations by augmenting network states with their associated flow counting processes and deriving Markov couplings of the augmented state-flow processes.
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    • "They are derived by using the so-called the precedence relation method. This is a systematic approach for the construction of bound models, which has been developed in [14] [15]. "
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    ABSTRACT: In this paper we study a production system consisting of a group of parallel machines producing multiple job types. Each machine has its own queue and it can process a restricted set of job types only. On arrival a job joins the shortest queue among all queues capable of serving that job. Under the assumption of Poisson arrivals and identical exponential processing times we derive upper and lower bounds for the mean waiting time. These bounds are obtained from so-called flexible bound models, and they provide a powerful tool to efficiently determine the mean waiting time. The bounds are used to study how the mean waiting time depends on the amount of overlap (i.e. common job types) between the machines.
    Operations Research-Spektrum 07/2001; 23(3):411-427. DOI:10.1007/PL00013360 · 0.99 Impact Factor
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    • "To see that (4) indeed is more general than the precedence relation method (Van Houtum et al. [24]), let α = δ x and β = "
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    ABSTRACT: Numerical methods for solving Markov chains are in general inefficient if the state space of the chain is very large (or infinite) and lacking a simple repeating structure. One alternative to solving such chains is to construct models that are simple to analyze and that provide bounds for a reward function of interest. We present a new bounding method for Markov chains inspired by Markov reward theory; our method constructs bounds by redirecting selected sets of transitions, facilitating an intuitive interpretation of the modifications on the original system. We show that our method is compatible with strong aggregation of Markov chains; thus we can obtain bounds for the initial chain by analyzing a much smaller chain. We illustrate our method on a problem of order fill rates for an inventory system of service tools.
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