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.

The shortest queue problem

1985 ◽  
Vol 22 (04) ◽  
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.

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.

scholarly journals On the Nonsymmetric Longer Queue Model: Joint Distribution, Asymptotic Properties, and Heavy Traffic Limits

2013 ◽  
Vol 2013 ◽  
pp. 1-21 ◽  
Author(s):  
Charles Knessl ◽  
Haishen Yao
Keyword(s):  
Heavy Traffic ◽  
Asymptotic Properties ◽  
Joint Probability ◽  
Integral Representations ◽  
Joint Probability Distribution ◽  
Single Server ◽  
Poisson Arrival ◽  
Large Numbers ◽  
Node Network ◽  
Number Of Customers

We consider two parallel queues, each with independent Poisson arrival rates, that are tended by a single server. The exponential server devotes all of its capacity to the longer of the queues. If both queues are of equal length, the server devotesνof its capacity to the first queue and the remaining1−νto the second. We obtain exact integral representations for the joint probability distribution of the number of customers in this two-node network. Then we evaluate this distribution in various asymptotic limits, such as large numbers of customers in either/both of the queues, light traffic where arrivals are infrequent, and heavy traffic where the system is nearly unstable.

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.

scholarly journals Heavy-traffic approx­imations for a layered network with limited resources

2018 ◽  
Vol 37 (2) ◽  
pp. 498-532
Author(s):  
Angelos Aveklouris ◽  
Maria Vlasiou ◽  
Jiheng Zhang ◽  
Bert Zwart
Keyword(s):  
Resource Sharing ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Limited Resources ◽  
Single Server ◽  
Processor Sharing ◽  
Bottleneck Node ◽  
General Distributions ◽  
Heavy Traffic Approximations ◽  
Number Of Customers

HEAVY-TRAFFIC APPROXIMATIONS FOR A LAYERED NETWORK WITH LIMITED RESOURCESMotivated by a web-server model, we present a queueing network consisting of two layers. The first layer incorporates the arrival of customers at a network of two single-server nodes. We assume that the interarrival and the service times have general distributions. Customers are served according to their arrival order at each node and after finishing their service they can re-enter at nodes several times for another service. At the second layer, active servers act as jobs that are served by a single server working at speed one in a processor-sharing fashion. We further assume that the degree of resource sharing is limited by choice, leading to a limited processor-sharing discipline. Our main result is a diffusion approximation for the process describing the number of customers in the system. Assuming a single bottleneck node and studying the system as it approaches heavy traffic, we prove a state-space collapse property.

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 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.

scholarly journals A New Sustainable Warehouse Management Approach for Workforce and Activities Scheduling

2021 ◽  
Vol 13 (4) ◽  
pp. 2021
Author(s):  
Vlado Popović ◽  
Milorad Kilibarda ◽  
Milan Andrejić ◽  
Borut Jereb ◽  
Dejan Dragan
Keyword(s):  
Linear Programming ◽  
Management Approach ◽  
Full Time ◽  
Warehouse Management ◽  
Logistics Systems ◽  
Positive Effects ◽  
Engineering Managers ◽  
Real Distribution ◽  
Key Factor ◽  
Number Of Customers

Sustainable engineering is very important for logistics systems. Nowadays, sustainable warehouse management is a key factor in market success. Workforce fluctuation and inverting the number of customers’ demands make a lot of problems in distribution warehouses. This study addresses a sustainable approach for the workforce scheduling problem recognized in a real distribution warehouse. The problem arises from the high variability of demand for workers over one workday, which causes workforce surplus in some periods of the workday and shortages in others. Engineering managers of the distribution warehouse already use different full-time and part-time shifts, and schedule workers on different activities, but they still have significant workforce surpluses or shortages in some periods. This study proposes the scheduling of activities’ execution together with workers to face that variability and decrease the cost of the workforce. This idea comes from the fact that some activities in a distribution warehouse can be done in a specific time period after the need for them occurs. In this way, the variability of demand for workers can be decreased, and a lower workforce cost may be ensured. Based on this idea, the entire problem is modeled as integer linear programming. The real example of the problem is solved, and the proposed model is tested on randomly generated instances of the problem in Python by means of the PuLP linear programming package. The results indicate different positive effects in the manner of sustainable warehouse management: lower workforce costs, time savings, better utilization of all types of resources and equipment, increased employee satisfaction, and so on. For even 61% of instances of the introduced problem, the obtained cost of the workforce is lower by more than 20% if activities’ executions are scheduled together with employees.

Linear Programming Techniques for the Construction of Palatable Human Diets

1994 ◽  
Vol 45 (5) ◽  
pp. 489-496 ◽  
Author(s):  
L. R. Fletcher ◽  
P. M. Soden ◽  
A. S. I. Zinober
Keyword(s):  
Linear Programming ◽  
Programming Techniques

An Integer Linear Programming-Based Method for the Extraction of Ontology Alignment

2021 ◽  
Vol 16 (2) ◽  
pp. 25-44
Author(s):  
Naima El Ghandour ◽  
Moussa Benaissa ◽  
Yahia Lebbah
Keyword(s):  
Linear Programming ◽  
Semantic Web ◽  
Integer Linear Programming ◽  
New Method ◽  
Ontology Matching ◽  
Ontology Alignment ◽  
Programming Techniques ◽  
Data Heterogeneity ◽  
Series Of Experiments

The Semantic Web uses ontologies to cope with the data heterogeneity problem. However, ontologies become themselves heterogeneous; this heterogeneity may occur at the syntactic, terminological, conceptual, and semantic levels. To solve this problem, alignments between entities of ontologies must be identified. This process is called ontology matching. In this paper, the authors propose a new method to extract alignment with multiple cardinalities using integer linear programming techniques. The authors conducted a series of experiments and compared them with currently used methods. The obtained results show the efficiency of the proposed method.

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