Two queues in parallel

Flatto, Leopold; McKean, H. P.

doi:10.1002/cpa.3160300206

Cited by 140 publications

(127 citation statements)

References 1 publication

Supporting

Mentioning

127

Contrasting

Order By: Relevance

“…(i) There exists exactly one product form am~n, 0< 1al < I and 0< 1131 <1, which satisfies the equilibrium equations (8) and (9). The factors a and 13 of this product form are equal to ql,-l…”

Section: The Compensation Approachmentioning

confidence: 99%

“…Part (ii) can be proved along the same lines. Let am~n, 0< Ial < 1 and 0< I~1< 1, be a solution of (8) and (9). Substitution of am~n in (8) and (9) gives qa~= ql,_1~2+q_l,la2+qO,_la~2+q_l.Oa2~+q_l,_la2~2 ,…”

Section: The Compensation Approachmentioning

confidence: 99%

“…He obtains the equilibrium joint queue length distribution at the two output buffers (as seen at the end of a clock cycle). He applies a uniformization technique that was used by Hatto and McKean [9] (see also Kingman [13]) for the shortest queue problem. Jaffe's results in [12] are presented in the form of a bivariate generating function.…”

Section: Introductionmentioning

confidence: 99%

“…A few two-dimensional queueing problems have recently yielded to an exact analysis, that is usually based on complex function theory. An often studied example of the latter type is the shortest queue problem, that was originally solved by Kingman [13] and Flatto and McKean [9], and that was later analyzed (in the asymmetric case) using the boundary value method [7]. The boundary value method aims to translate the problem of determining the generating function of the joint distribution of the random walk/queueing problem into a boundary value problem of Dirichlet, Riemann or Riemann-Hilbert type.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Compensation Approach Applied to a 2 × 2 Switch

Boxma

Houtum

1993

Prob. Eng. Inf. Sci.

View full text Add to dashboard Cite

In this paper we analyze an asymmetric 2 x 2 buffered switch, fed by two independent Bernoulli input streams. We derive the joint equilibrium distribution of the numbers of messages waiting in the two output buffers. This joint distribution is presented explicitly, without the use of generating functions, in the form of a sum of two alternating series of product-form geometric distributions. The method used is the so-called compensation approach, developed by Adan,Wessels and Zijm.

show abstract

Section: The Compensation Approachmentioning

confidence: 99%

Section: The Compensation Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Compensation Approach Applied to a 2 × 2 Switch

Boxma

Houtum

1993

Prob. Eng. Inf. Sci.

View full text Add to dashboard Cite

show abstract

“…The shortest queue problem was formulated originally by Haight in [12], however, the first major step towards its analysis was made by Kingman in [14] and Flatto and McKean in [10]. Using a uniformization approach they proved that the generating function of the equilibrium distribution is meromorphic and they established explicit relations for the poles and residues.…”

Section: Introductionmentioning

confidence: 99%

Shortest expected delay routing for Erlang servers

Adan

Wessels

1996

Queueing Syst

View full text Add to dashboard Cite

The queueing problem with a Poisson arrival stream and two identical Erlang servers is analysed for the queueing discipline based on shortest expected delay. This queueing problem may be represented as a random walk on the integer grid in the first quadrant of the plane. In the paper it is shown that the equilibrium distribution of this random walk can be written as a countable linear combination of product forms. This linear combination is constructed in a compensation procedure. In this case the compensation procedure is essentially more complicated than in other cases where the same idea was exploited. The reason for the complications is that in this case the boundary consists of several layers which in turn is caused by the fact that transitions starting in inner states are not restricted to end in neighbouring states.Good starting solutions for the compensation procedure are found by solving the shortest expected delay problem with the same service distributions but with instantaneous jockeying.It is also shown that the results can be used for an efficient computation of relevant performance criteria.

show abstract

Algorithmic and approximation analyses of the shorter queue model

Rao

Posner

1987

Naval Research Logistics

View full text Add to dashboard Cite

A system of two parallel queues where the arrivals from a single stream of customers join the shorter queue is considered. Arrivals form a homogeneous Poisson stream and the service times in each of the two queues are independent exponential variates. By treating one of the queues as bounded, the steady‐state probability vector for the system can be expressed in a modified matrix‐geometric form and can be computed efficiently. Computational procedures for the sojourn time distribution and characteristics of the departure stream are developed. Some numerical results are presented, and based on these results an efficient approximation scheme for the model is developed which can be readily extended to systems with more than two parallel queues.

show abstract

Two queues in parallel

Cited by 140 publications

References 1 publication

The Compensation Approach Applied to a 2 × 2 Switch

The Compensation Approach Applied to a 2 × 2 Switch

Shortest expected delay routing for Erlang servers

Algorithmic and approximation analyses of the shorter queue model

Contact Info

Product

Resources

About