Exact buffer overflow calculations for queues via martingales

Søren Asmussen; Manfred Jobmann; Hans Peter Schwefel

doi:10.1023/A:1019994728099

Exact buffer overflow calculations for queues via martingales

Søren Asmussen^*, Manfred Jobmann, Hans Peter Schwefel

^*Corresponding author for this work

Research output: Contribution to journal › Journal article › Research › peer-review

20 Citations (Scopus)

Abstract

Let τ_n be the first time a queueing process like the queue length or workload exceeds a level n. For the M/M/1 queue length process, the mean double struck E sugneτ_n and the Laplace transform double-struck E signe^-Sτn is derived in closed form using a martingale introduced in Kella and Whitt (1992). For workload processes and more general systems like MAP/PH/1, we use a Markov additive extension given in Asmussen and Kella (2000) to derive sets of linear equations determining the same quantities. Numerical illustrations are presented in the framework of M/M/ 1 and MMPP/M/1 with an application to performance evaluation of telecommunication systems with long-range dependent properties in the packet arrival process. Different approximations that are obtained from asymptotic theory are compared with exact numerical results.

Original language	English
Journal	Queueing Systems
Volume	42
Issue number	1
Pages (from-to)	63-90
Number of pages	28
ISSN	0257-0130
DOIs	https://doi.org/10.1023/A:1019994728099
Publication status	Published - 1 Dec 2002
Externally published	Yes

Keywords

Exponential martingale
Extreme value theory
Lévy process
Local time
Markov-modulation
Martingale
Power tail
Queue length
Regenerative process
Wald martingale

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title = "Exact buffer overflow calculations for queues via martingales",

abstract = "Let τn be the first time a queueing process like the queue length or workload exceeds a level n. For the M/M/1 queue length process, the mean double struck E sugneτn and the Laplace transform double-struck E signe-Sτn is derived in closed form using a martingale introduced in Kella and Whitt (1992). For workload processes and more general systems like MAP/PH/1, we use a Markov additive extension given in Asmussen and Kella (2000) to derive sets of linear equations determining the same quantities. Numerical illustrations are presented in the framework of M/M/ 1 and MMPP/M/1 with an application to performance evaluation of telecommunication systems with long-range dependent properties in the packet arrival process. Different approximations that are obtained from asymptotic theory are compared with exact numerical results.",

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AU - Jobmann, Manfred

AU - Schwefel, Hans Peter

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N2 - Let τn be the first time a queueing process like the queue length or workload exceeds a level n. For the M/M/1 queue length process, the mean double struck E sugneτn and the Laplace transform double-struck E signe-Sτn is derived in closed form using a martingale introduced in Kella and Whitt (1992). For workload processes and more general systems like MAP/PH/1, we use a Markov additive extension given in Asmussen and Kella (2000) to derive sets of linear equations determining the same quantities. Numerical illustrations are presented in the framework of M/M/ 1 and MMPP/M/1 with an application to performance evaluation of telecommunication systems with long-range dependent properties in the packet arrival process. Different approximations that are obtained from asymptotic theory are compared with exact numerical results.

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KW - Exponential martingale

KW - Extreme value theory

KW - Lévy process

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KW - Martingale

KW - Power tail

KW - Queue length

KW - Regenerative process

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Exact buffer overflow calculations for queues via martingales

Abstract

Keywords

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