Analysis of Queues with Markovian Service Processes

Ozawa, Toshihisa

doi:10.1081/stm-200033073

Cited by 18 publications

(16 citation statements)

References 21 publications

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“…Queues with variable service and arrival speeds arise naturally in practice and therefore many classical works can be found. Most of the results deal with the single server queue, see for example Takine (2005), Ozawa (2004), Sengupta (1990) and references therein. Neuts (1981) analyzed the M/M/1 queue as well as the M/M/C queue in random environment by using the matrix-geometric approach while Takine and Sengupta (1997) looked at the infinite server queue when only the arrival process was subject to a Markovian modulation.…”

Section: Introductionmentioning

confidence: 99%

M/M/∞ queues in semi-Markovian random environment

D’Auria

2008

Queueing Syst

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Abstract. In this paper we investigate an M/M/∞ queue whose parameters depend on an external random environment that we assume to be a semi-Markovian process with finite state space. For this model we show a recursive formula that allows to compute all the factorial moments for the number of customers in the system in steady state. The used technique is based on the calculation of the raw moments of the measure of a bidimensional random set. Finally the case when the random environment has only two states is deeper analyzed. We obtain an explicit formula to compute the above mentioned factorial moments when at least one of the two states has sojourn time exponentially distributed.

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Section: Introductionmentioning

confidence: 99%

M/M/∞ queues in semi-Markovian random environment

D’Auria

2008

Queueing Syst

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“…Of course, the stationary waiting time distribution of a GI/N/1 queue is matrix exponential [13,14], but the class of queueing models defined by QBD processes includes not only MAP/PH/1 queues but also Markovian queueing models in which arrival and service processes are mutually dependent. One example of those Markovian queueing models is a MAP/MSP/1 queue [11], where MSP is the abbreviation of Markovian service process; the MSP is a model similar to a Markovian arrival process (MAP) [7,8], where arrivals are replaced with service completions. The MSP can represent various queueing models such as vacation models, N -policy models, and exceptional service models.…”

Section: Introductionmentioning

confidence: 99%

Sojourn time distributions in the queue defined by a general QBD process

Ozawa

2006

Queueing Syst

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We consider a general QBD process as defining a FIFO queue and obtain the stationary distribution of the sojourn time of a customer in that queue as a matrix exponential distribution, which is identical to a phase-type distribution under a certain condition. Since QBD processes include many queueing models where the arrival and service process are dependent, these results form a substantial generalization of analogous results reported in the literature for queues such as the PH/PH/c queue. We also discuss asymptotic properties of the sojourn time distribution through its matrix exponential form.

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“…We note that the Lindley equation (1) perfectly describes the queue waiting time in a queueing system with renewal-type services using the first come first serve (FCFS) service discipline. However, the situation is different when the service times are of the more general Markov renewal type in which case the Lindley recurrence may come short of accommodating different models based on how the service process behaves when the queue is empty [7]. In the current paper, we focus on queues that are described solely by the Lindley equation (1) as in [6].…”

Section: Introductionmentioning

confidence: 99%

“…In the current paper, we focus on queues that are described solely by the Lindley equation (1) as in [6]. For the case of non-renewal service times, more general models, for instance the ones discussed in [7], with more general descriptions than that of the recurrence (1), are left for future research. Queueing systems with arrivals governed by a semi-Markov process have been studied extensively since an SM/M/1 queue was first analyzed in [8].…”

Section: Introductionmentioning

confidence: 99%

“…Studies related to auto-correlated service times are less common and we now list a few from the existing literature. A MAP/MSP/1 queue is studied in [7] using matrix-analytical techniques in terms of the queue length, but waiting time results are not given. The Reference [21] studies a queue whose service speed changes according to an external environment governed by a Markov process.…”

Section: Introductionmentioning

confidence: 99%

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System-theoretical algorithmic solution to waiting times in semi-Markov queues

Akar¹,

Sohraby²

2009

Performance Evaluation

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a b s t r a c tMarkov renewal processes with matrix-exponential semi-Markov kernels provide a generic tool for modeling auto-correlated interarrival and service times in queueing systems. In this paper, we study the steady-state actual waiting time distribution in an infinite capacity single-server semi-Markov queue with the auto-correlation in interarrival and service times modeled by Markov renewal processes with matrix-exponential kernels. Our approach is based on the equivalence between the waiting time distribution of this semi-Markov queue and the output of a linear feedback interconnection system. The unknown parameters of the latter system need to be determined through the solution of a SDC (Spectral-Divide-and-Conquer) problem for which we propose to use the ordered Schur decomposition. This approach leads us to a completely matrix-analytical algorithm to calculate the steady-state waiting time which has a matrix-exponential distribution. Besides its unifying structure, the proposed algorithm is easy to implement and is computationally efficient and stable. We validate the effectiveness and the generality of the proposed approach through numerical examples.

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Analysis of Queues with Markovian Service Processes

Cited by 18 publications

References 21 publications

M/M/∞ queues in semi-Markovian random environment

M/M/∞ queues in semi-Markovian random environment

Sojourn time distributions in the queue defined by a general QBD process

System-theoretical algorithmic solution to waiting times in semi-Markov queues

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