ON THE INFINITE SERVER SHORTEST QUEUE PROBLEM: SYMMETRIC CASE

2005 ◽  
Vol 21 (1) ◽  
pp. 101-132 ◽  
Author(s):  
Haishen Yao ◽  
Charles Knessl
Keyword(s):  
Symmetric Case ◽  
Infinite Server ◽  
Shortest Queue ◽  
Shortest Queue Problem

The Equilibrium Distribution for a Clocked Buffered Switch

1992 ◽  
Vol 6 (4) ◽  
pp. 425-438 ◽  
Author(s):  
Steven Jaffe
Keyword(s):  
Joint Distribution ◽  
Equilibrium Distribution ◽  
Heavy Traffic ◽  
Basic Element ◽  
Asymptotic Results ◽  
Heavy Traffic Limit ◽  
Parallel Data ◽  
Shortest Queue ◽  
Bivariate Generating Function ◽  
Shortest Queue Problem

A 2-by-2 buffered switch is the basic element of certain parallel data-processing networks. For a switch fed by two independent Bernoulli input streams, we find the joint distribution of the number of messages waiting in the two buffers at equilibrium, in the form of a bivariate generating function. The derivation uses complex-variable techniques developed by Kingman and by Flatto and McKean for the “shortest queue problem.” A number of asymptotic results are given, the principal one being the variance of the total number of waiting messages in the heavy-traffic limit.

A new heavy traffic limit for the asymmetric shortest queue problem

1999 ◽  
Vol 10 (5) ◽  
pp. 497-509 ◽  
Author(s):  
CHARLES KNESSL
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Parabolic Partial Differential Equation ◽  
Service Rate ◽  
Heavy Traffic Limit ◽  
Dimensional Approximation ◽  
Shortest Queue ◽  
Shortest Queue Problem

We consider the classic shortest queue problem in the heavy traffic limit. We assume that the second server works slowly and that the service rate of the first server is nearly equal to the arrival rate. Solving for the (asymptotic) joint steady state queue length distribution involves analyzing a backward parabolic partial differential equation, together with appropriate side conditions. We explicitly solve this problem. We thus obtain a two-dimensional approximation for the steady state queue length probabilities.

The shortest queue problem

1985 ◽  
Vol 22 (4) ◽  
pp. 865-878 ◽  
Author(s):  
Shlomo Halfin
Keyword(s):  
Linear Programming ◽  
Heavy Traffic ◽  
Single Server ◽  
Service Center ◽  
Poisson Stream ◽  
Programming Techniques ◽  
Shortest Queue ◽  
Number Of Customers ◽  
Queues In Parallel ◽  
Shortest Queue Problem

A Poisson stream of customers arrives at a service center which consists of two single-server queues in parallel. The service times of the customers are exponentially distributed, and both servers serve at the same rate. Arriving customers join the shortest of the two queues, with ties broken in any plausible manner. No jockeying between the queues is allowed. Employing linear programming techniques, we calculate bounds for the probability distribution of the number of customers in the system, and its expected value in equilibrium. The bounds are asymptotically tight in heavy traffic.

scholarly journals Bounding functions of Markov processes and the shortest queue problem

1989 ◽  
Vol 21 (4) ◽  
pp. 842-860
Author(s):  
John A. Gubner ◽  
B. Gopinath ◽  
S. R. S. Varadhan
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Heavy Traffic ◽  
Stable State ◽  
Homogeneous Markov Process ◽  
Negative Function ◽  
Shortest Queue ◽  
The Stability ◽  
Uniformly Bounded ◽  
Shortest Queue Problem

We prove a theorem which can be used to show that the expectation of a non-negative function of the state of a time-homogeneous Markov process is uniformly bounded in time. This is reminiscent of the classical theory of non-negative supermartingales, except that our analog of the supermartingale inequality need not hold almost surely. Consequently, the theorem is suitable for establishing the stability of systems that evolve in a stabilizing mode in most states, though from certain states they may jump to a less stable state. We use this theorem to show that ‘joining the shortest queue' can bound the expected sum of the squares of the differences between all pairs among N queues, even under arbitrarily heavy traffic.

scholarly journals Analysis of the asymmetric shortest queue problem

1991 ◽  
Vol 8 (1) ◽  
pp. 1-58 ◽  
Author(s):  
I. J. B. F. Adan ◽  
J. Wessels ◽  
W. H. M. Zijm
Keyword(s):  
Shortest Queue ◽  
Shortest Queue Problem

Analysis of the asymmetric shortest queue problem with threshold jockeying

1991 ◽  
Vol 7 (4) ◽  
pp. 615-627 ◽  
Author(s):  
I. J. B. F. Adan ◽  
J. WESSELS ◽  
W. H. M. Zijm.
Keyword(s):  
Shortest Queue ◽  
Shortest Queue Problem

scholarly journals Bounding functions of Markov processes and the shortest queue problem

1989 ◽  
Vol 21 (04) ◽  
pp. 842-860
Author(s):  
John A. Gubner ◽  
B. Gopinath ◽  
S. R. S. Varadhan
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Heavy Traffic ◽  
Stable State ◽  
Homogeneous Markov Process ◽  
Negative Function ◽  
Shortest Queue ◽  
The Stability ◽  
Uniformly Bounded ◽  
Shortest Queue Problem

We prove a theorem which can be used to show that the expectation of a non-negative function of the state of a time-homogeneous Markov process is uniformly bounded in time. This is reminiscent of the classical theory of non-negative supermartingales, except that our analog of the supermartingale inequality need not hold almost surely. Consequently, the theorem is suitable for establishing the stability of systems that evolve in a stabilizing mode in most states, though from certain states they may jump to a less stable state. We use this theorem to show that ‘joining the shortest queue' can bound the expected sum of the squares of the differences between all pairs among N queues, even under arbitrarily heavy traffic.

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