Waiting time distributions for M/M/N processor sharing queues

Braband, Jens

doi:10.1080/15326349408807309

Cited by 13 publications

(16 citation statements)

References 2 publications

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“…[15,19], in the case of ϕ r (n) = min(r/n, 1), n ∈ N, an r-server PS system, where all requests are served in a PS mode, but each request receives at most the capacity of one processor, cf. [8, p. 283], [3,4,9], in case of ϕ r,k (n) = min(r/(n + k), 1), n ∈ N, an r-server PS system with k ∈ N permanent requests, in the case of ϕ ∞ (n) = 1, n ∈ N, an infinite-server system.…”

Section: Introductionmentioning

confidence: 99%

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On sojourn times in M/GI systems under state-dependent processor sharing

Brandt

2009

Queueing Syst

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We consider a system with Poisson arrivals and i.i.d. service times. The requests are served according to the state-dependent processor-sharing discipline, where each request receives a service capacity which depends on the actual number of requests in the system. The linear systems of PDEs describing the residual and attained sojourn times coincide for this system, which provides time reversibility including sojourn times for this system, and their minimal non-negative solution gives the LST of the sojourn time V (τ ) of a request with required service time τ . For the case that the service time distribution is exponential in a neighborhood of zero, we derive a linear system of ODEs, whose minimal non-negative solution gives the LST of V (τ ), and which yields linear systems of ODEs for the moments of V (τ ) in the considered neighborhood of zero. Numerical results are presented for the variance of V (τ ). In the case of an M/GI /2-PS system, the LST of V (τ ) is given in terms of the solution of a convolution equation in the considered neighborhood of zero. For service times bounded from below, surprisingly simple expressions for the LST and variance of V (τ ) in this neighborhood of zero are derived, which yield in particular the LST and variance of V (τ ) in M/D/2-PS.Keywords Poisson arrivals · General service times · Locally exponential service times · Deterministic service times · State-dependent processor sharing · Generalized processor sharing · Many-server · Time reversibility · Insensitivity · M/GI /r-PS · M/D/r-PS · M/D/2-PS · Sojourn time · Laplace-Stieltjes transform · Moments M. Brandt ( ) Konrad-

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Section: Introductionmentioning

confidence: 99%

“…The variance of V (τ ) in the M/M/2-PS system is given in [14]. The Laplace-Stieltjes transform (LST) and moments of V (τ ) in the general M/M/r-PS system are treated in [3] and in the M/M/SDPS system in [4]. The aim of this paper is to derive analytical results and representations for sojourn times in the M/GI /SDPS system.…”

Section: Introductionmentioning

confidence: 99%

On sojourn times in M/GI systems under state-dependent processor sharing

Brandt

2009

Queueing Syst

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“…[28,32]; in case of ϕ m (n) = min(m/n, 1), n ∈ N, an m-server PS system, where all requests are served in a PS mode but each request receives at most the service capacity of one processor, cf. [14, p. 283], [6,7,16]; in case of ϕ m,k (n) = min(m/(n + k), 1), n ∈ N, an m-server PS system with k ∈ N permanent requests.…”

Section: Introductionmentioning

confidence: 99%

“…The general M/M/m − PS system is treated in [6,7]. By using the approach of [13], a system of differential equations is derived for the LSTs of the conditional waiting times W n (τ ) of a request with required service time τ and which finds n requests at its arrival in the system.…”

Section: Introductionmentioning

confidence: 99%

Waiting times for M/M systems under state-dependent processor sharing

Brandt

2008

Queueing Syst

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We consider a system where the arrivals form a Poisson process and the required service times of the requests are exponentially distributed. The requests are served according to the state-dependent (Cohen's generalized) processor sharing discipline, where each request in the system receives a service capacity which depends on the actual number of requests in the system. For this system we derive systems of ordinary differential equations for the LST and for the moments of the conditional waiting time of a request with given required service time as well as a stable and fast recursive algorithm for the LST of the second moment of the conditional waiting time, which in particular yields the second moment of the unconditional waiting time. Moreover, asymptotically tight upper bounds for the moments of the conditional waiting time are given. The presented numerical results for the first two moments of the sojourn times in M/M/m − PS systems show that the proposed algorithms work well.

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“…In the case in which ϕ m (n) = min(m/n, 1), n ∈ N, we have an M/GI/m-PS system; i.e. an m-server PS system, where all requests are served in a PS mode, but each request receives at most the capacity of one processor; see [11, p. 283], [5], and [12]. In the case in which ϕ m,k (n) = min(m/(n + k), 1), n ∈ N, we have an m-server PS system with k ∈ N permanent requests.…”

Section: ) If There Arementioning

confidence: 99%

Insensitive Bounds for the Moments of the Sojourn Times in M/GI Systems Under State-Dependent Processor Sharing

Brandt¹,

Brandt²

2010

Adv. Appl. Probab.

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We consider a system with Poisson arrivals and independent and identically distributed service times, where requests in the system are served according to the state-dependent (Cohen's generalized) processor-sharing discipline, where each request receives a service capacity that depends on the actual number of requests in the system. For this system, we derive expressions as well as tight insensitive upper bounds for the moments of the conditional sojourn time of a request with given required service time. The bounds generalize and extend corresponding results, recently given for the single-server processor-sharing system in Cheung et al.

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Waiting time distributions for M/M/N processor sharing queues

Cited by 13 publications

References 2 publications

On sojourn times in M/GI systems under state-dependent processor sharing

On sojourn times in M/GI systems under state-dependent processor sharing

Waiting times for M/M systems under state-dependent processor sharing

Insensitive Bounds for the Moments of the Sojourn Times in M/GI Systems Under State-Dependent Processor Sharing

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