Application of level crossing analysis to discrete state processes in queueing systems

Shanthikumar, J. George; Chandra, M. Jeya

doi:10.1002/nav.3800290407

Cited by 20 publications

(4 citation statements)

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“…The system performance measures were numerically evaluated using an algorithmic approach proposed by Neuts [9] and a discrete state level crossing analysis (see [11]). …”

Section: Resultsmentioning

confidence: 99%

An Algorithmic Solution for a Queueing Model of a Computer System with Interactive and Batch Jobs

Shanthikumar¹,

Sargent²

1981

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“…The system performance measures were numerically evaluated using an algorithmic approach proposed by Neuts [9] and a discrete state level crossing analysis (see [11]). …”

Section: Resultsmentioning

confidence: 99%

An Algorithmic Solution for a Queueing Model of a Computer System with Interactive and Batch Jobs

Shanthikumar¹,

Sargent²

1981

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“…Proof: Define In steady state, let IVLt and iVb be the total number of steps in the arrival and departure processes, respectively, over a given,long period of time. Using the level crossing method due to Foster and Perera [10] or Shanthikumqr and Chandra [17], in the steady state, the nurnber of times a given level is crossed in the arrlval process can differ at most by one from the number of times it is crossed in the departure or service process, i.e. It is the product of the p.a.e.…”

Section: This Is Explainedmentioning

confidence: 99%

Computational Aspects of Bulk-Service Queueing System With Variable Capacity and Finite Waiting Space: M/G_y/1/N+b

Chaudhry

Gupta

Madill³

1991

JORSJ

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“…Furthennore, Eqs. (65a) and (65b) can be derived directly using the discrete state-level crossing analysis (e.g., see [25] for a detailed discussion on discrete state-level crossing analysis). Let Q(r) be the number of "phases" in the system at time r. Each customer on its arrival brings a number of phases distributed according to g. Each phase requires an exponentially distributed service with mean 1 /p.…”

Section: Letmentioning

confidence: 99%

“…The lemma now follows from Eqs (23). and(25). The closure property of BHP distribution under convolution now easily follows from Eq.…”

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confidence: 93%

Bilateral phase‐type distributions

Shanthikumar

1985

Naval Research Logistics

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In this article we define a class of distributions called bilateral phase type (BPH), and study its closure and computational properties. The class of BPH distributions is closed under convolution, negative convolution, and mixtures. The one-sided version of BPH, called generalized phase type (GPH), is also defined. The class of GPH distributions is strictly larger than the class of phase-type distributions introduced by Neuts, and is closed under convolution, negative convolution with nonnegativity condition, mixtures, and formation of cohere'nt systems. We give computational schemes to compute the resulting distributions from the above operations and extend them to analyze queueing processes. In particular, we present efficient algorithms to compute the steady-state and transient waiting times in GPH/GPH/ 1 queues and a simple algorithm to compute the steady-state waiting time in M/GPH/ 1 queues. INTRODUCTIONNeuts [ 171, using the properties of exponential, generalized Erlang, and hyperexponential distributions as a base, developed a phase-type probability distribution function. Through a sequence of several articles, he has established its closure and computational properties, which make its use very attractive in applied probability modeling (see [20] for a recent account of these results). The purpose of this article is to present two classes of distributions, called bilateral phase type and generalized phase type, and to establish their closure and computational properties. The class of generalized phase-type distributions is strictly larger than the class of phase-type distributions.The phase-type distribution of Neuts can be interpreted as a random sum of independent exponential random variables via the uniformization procedure [9; 18,p.78;23;24]. The stopping time for this sum has an associated discrete phase-type distribution. A natural extension of the phase-type distribution is to consider the sum of independent and identically distributed (iid) exponential random variables (rv) over an arbitrary random number. We will call such a sum a generalized phase-type random variable. Let (g(n))," be the probability distribution function of a random number L , and define where (En): are iid exponential rv's with mean l / A and Eo = 0 with probability (wp) 1. The survival function Fx of X is then *Present address:

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Application of level crossing analysis to discrete state processes in queueing systems

Cited by 20 publications

References 21 publications

An Algorithmic Solution for a Queueing Model of a Computer System with Interactive and Batch Jobs

An Algorithmic Solution for a Queueing Model of a Computer System with Interactive and Batch Jobs

Computational Aspects of Bulk-Service Queueing System With Variable Capacity and Finite Waiting Space: M/G_y/1/N+b

Bilateral phase‐type distributions

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