The autostrada queueing problem

The model considered in this note has been referred to by Haight (1958), Kingman (1961) and Flatto and McKean (1977) as two queues in parallel. Customers choose the shorter of the two queues which are otherwise independent. This system is known to be inferior to a single queue feeding the two servers, but how much? Some elementary considerations provide a fresh perspective on this awkward boundary-value problem. A procedure is proposed for the solution in the context of finite waiting-room size and some comparisons are made with the single-queue system and an independent two-queue system.

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Workload analysis of a two-queue system by formulating a boundary value problem

Modelling and Performance Evaluation Methodology - Lecture Notes in Control and Information Sciences ◽

10.1007/bfb0005190 ◽

2006 ◽

pp. 518-544

Author(s):

Philippe Nain

Keyword(s):

Boundary Value Problem ◽

Boundary Value ◽

Workload Analysis ◽

Queue System

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A two-point Markov chain boundary-value problem

Advances in Applied Probability ◽

10.2307/1427526 ◽

1993 ◽

Vol 25 (3) ◽

pp. 607-630 ◽

Cited By ~ 10

Author(s):

D. J. Daley ◽

L. D. Servi

Keyword(s):

Markov Chain ◽

Boundary Value Problem ◽

Boundary Value ◽

Sample Path ◽

Maximum Likelihood Estimates ◽

Initial Time ◽

Time Interval ◽

Birth Process ◽

Waiting Room ◽

Time Point

The two-point Markov chain boundary-value problem discussed in this paper is a finite-time version of the quasi-stationary behaviour of Markov chains. Specifically, for a Markov chain {Xt:t = 0, 1, ·· ·}, given the time interval (0, n), the interest is in describing the chain at some intermediate time point r conditional on knowing both the behaviour of the chain at the initial time point 0 and that over the interval (0, n) it has avoided some subset B of the state space. The paper considers both ‘real time' estimates for r = n (i.e. the chain has avoided B since 0), and a posteriori estimates for r < n with at least partial knowledge of the behaviour of Xn. Algorithms to evaluate the distribution of Xr can be as small as O(n3) (and, for practical purposes, even O(n2 log n)). The estimates may be stochastically ordered, and the process (and hence, the estimates) may be spatially homogeneous in a certain sense. Maximum likelihood estimates of the sample path are furnished, but by example we note that these ML paths may differ markedly from the path consisting of the expected or average states. The scope for two-point boundary-value problems to have solutions in a Markovian setting is noted.Several examples are given, together with a discussion and examples of the analogous problem in continuous time. These examples include the basic M/G/k queue and variants that include a finite waiting room, reneging, balking, and Bernoulli feedback, a pure birth process and the Yule process. The queueing examples include Larson's (1990) ‘queue inference engine'.

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A two-point Markov chain boundary-value problem

Advances in Applied Probability ◽

10.1017/s000186780002557x ◽

1993 ◽

Vol 25 (03) ◽

pp. 607-630 ◽

Cited By ~ 2

Author(s):

D. J. Daley ◽

L. D. Servi

Keyword(s):

Markov Chain ◽

Boundary Value Problem ◽

Boundary Value ◽

Sample Path ◽

Maximum Likelihood Estimates ◽

Initial Time ◽

Time Interval ◽

Birth Process ◽

Waiting Room ◽

Time Point

The two-point Markov chain boundary-value problem discussed in this paper is a finite-time version of the quasi-stationary behaviour of Markov chains. Specifically, for a Markov chain {Xt :t = 0, 1, ·· ·}, given the time interval (0, n), the interest is in describing the chain at some intermediate time point r conditional on knowing both the behaviour of the chain at the initial time point 0 and that over the interval (0, n) it has avoided some subset B of the state space. The paper considers both ‘real time' estimates for r = n (i.e. the chain has avoided B since 0), and a posteriori estimates for r < n with at least partial knowledge of the behaviour of Xn. Algorithms to evaluate the distribution of Xr can be as small as O(n 3) (and, for practical purposes, even O(n 2 log n)). The estimates may be stochastically ordered, and the process (and hence, the estimates) may be spatially homogeneous in a certain sense. Maximum likelihood estimates of the sample path are furnished, but by example we note that these ML paths may differ markedly from the path consisting of the expected or average states. The scope for two-point boundary-value problems to have solutions in a Markovian setting is noted. Several examples are given, together with a discussion and examples of the analogous problem in continuous time. These examples include the basic M/G/k queue and variants that include a finite waiting room, reneging, balking, and Bernoulli feedback, a pure birth process and the Yule process. The queueing examples include Larson's (1990) ‘queue inference engine'.

Download Full-text