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 Heavy-Traffic Comparison of a Discrete-Time Generalized Processor Sharing Queue and a Pure Randomly Alternating Service Queue

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2723
Author(s):  
Arnaud Devos ◽  
Joris Walraevens ◽  
Dieter Fiems ◽  
Herwig Bruneel
Keyword(s):  
Functional Equation ◽  
Discrete Time ◽  
Heavy Traffic ◽  
The Other ◽  
Queueing Models ◽  
Single Server ◽  
Processor Sharing ◽  
Two Dimensional ◽  
Heavy Traffic Limit ◽  
Heavy Traffic Approximations

This paper compares two discrete-time single-server queueing models with two queues. In both models, the server is available to a queue with probability 1/2 at each service opportunity. Since obtaining easy-to-evaluate expressions for the joint moments is not feasible, we rely on a heavy-traffic limit approach. The correlation coefficient of the queue-contents is computed via the solution of a two-dimensional functional equation obtained by reducing it to a boundary value problem on a hyperbola. In most server-sharing models, it is assumed that the system is work-conserving in the sense that if one of the queues is empty, a customer of the other queue is served with probability 1. In our second model, we omit this work-conserving rule such that the server can be idle in case of a non-empty queue. Contrary to what we would expect, the resulting heavy-traffic approximations reveal that both models remain different for critically loaded queues.

Stochastic bounds for heterogeneous-server queues with Erlang service times

1974 ◽  
Vol 11 (04) ◽  
pp. 785-796 ◽  
Author(s):  
Oliver S. Yu
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Single Server ◽  
Shape Parameters ◽  
Server Queue ◽  
Service Times ◽  
Heavy Traffic Approximations ◽  
Multi Server

This paper establishes stochastic bounds for the phasal departure times of a heterogeneous-server queue with a recurrent input and Erlang service times. The multi-server queue is bounded by a simple GI/E/1 queue. When the shape parameters of the Erlang service-time distributions of different servers are the same, these relations yield two-sided bounds for customer waiting times and the queue length, which can in turn be used with known results for single-server queues to obtain characterizations of steady-state distributions and heavy-traffic approximations.

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.

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 Analysis of Closed Queueing Networks with Batch Service

2020 ◽  
Vol 20 (4) ◽  
pp. 527-533
Author(s):  
Elena P. Stankevich ◽  
◽  
Igor E. Tananko ◽  
Vitalii I. Dolgov ◽  
◽  
...  
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Queueing Networks ◽  
Manufacturing Systems ◽  
Queueing Network ◽  
Transport Systems ◽  
Batch Service ◽  
Single Server ◽  
Closed Queueing Networks ◽  
Number Of Customers

We consider a closed queuing network with batch service and movements of customers in continuous time. Each node in the queueing network is an infinite capacity single server queueing system under a RANDOM discipline. Customers move among the nodes following a routing matrix. Customers are served in batches of a fixed size. If a number of customers in a node is less than the size, the server of the system is idle until the required number of customers arrive at the node. An arriving at a node customer is placed in the queue if the server is busy. The batсh service time is exponentially distributed. After a batсh finishes its execution at a node, each customer of the batch, regardless of other customers of the batch, immediately moves to another node in accordance with the routing probability. This article presents an analysis of the queueing network using a Markov chain with continuous time. The qenerator matrix is constructed for the underlying Markov chain. We obtain expressions for the performance measures. Some numerical examples are provided. The results can be used for the performance analysis manufacturing systems, passenger and freight transport systems, as well as information and computing systems with parallel processing and transmission of information.

Stochastic bounds for heterogeneous-server queues with Erlang service times

1974 ◽  
Vol 11 (4) ◽  
pp. 785-796 ◽  
Author(s):  
Oliver S. Yu
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Single Server ◽  
Shape Parameters ◽  
Server Queue ◽  
Service Times ◽  
Heavy Traffic Approximations ◽  
Multi Server

This paper establishes stochastic bounds for the phasal departure times of a heterogeneous-server queue with a recurrent input and Erlang service times. The multi-server queue is bounded by a simple GI/E/1 queue. When the shape parameters of the Erlang service-time distributions of different servers are the same, these relations yield two-sided bounds for customer waiting times and the queue length, which can in turn be used with known results for single-server queues to obtain characterizations of steady-state distributions and heavy-traffic approximations.

Polling in a Closed Network

1994 ◽  
Vol 8 (3) ◽  
pp. 327-343 ◽  
Author(s):  
Eltan Altman ◽  
Uri Yechiali
Keyword(s):  
Dynamic Control ◽  
Probability Generating Function ◽  
Queueing Network ◽  
Arbitrary Point ◽  
Fixed Number ◽  
Service Discipline ◽  
Cycle Duration ◽  
Single Server ◽  
Optimal Dynamic ◽  
Number Of Customers

We consider a closed queueing network with a fixed number of customers, where a single server moves cyclically between N stations, rending service in each station according to some given discipline (Gated, Exhaustive, or the Globally Gated regime). When service of a customer (message) ends in station j, it is routed to station k with probability Pjk. We derive explicit expressions for the probability generating function and the moments of the number of customers at the various queues at polling instants and calculate the mean cycle duration and throughput for each service discipline. We then obtain the first moments of the queues' length at an arbitrary point in time. A few examples are given to illustrate the analysis. Finally, we address the problem of optimal dynamic control of the order of stations to be served.

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.

Heavy Traffic Approximations of a Queue with Varying Service Rates and General Arrivals

2012 ◽  
Vol 28 (1) ◽  
pp. 63-108 ◽  
Author(s):  
Robert Buche ◽  
Arka P. Ghosh ◽  
Vladas Pipiras
Keyword(s):  
Heavy Traffic ◽  
Service Rates ◽  
Heavy Traffic Approximations
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