A service model in which the server is required to search for customers

1984 ◽  
Vol 21 (1) ◽  
pp. 157-166 ◽  
Author(s):  
Marcel F. Neuts ◽  
M. F. Ramalhoto
Keyword(s):  
Poisson Process ◽  
Numerical Computation ◽  
Probability Distributions ◽  
General Distribution ◽  
Search Process ◽  
Single Server ◽  
Stationary Probability ◽  
Service Model ◽  
Service Times ◽  
Number Of Customers

Customers enter a pool according to a Poisson process and wait there to be found and processed by a single server. The service times of successive items are independent and have a common general distribution. Successive services are separated by seek phases during which the server searches for the next customer. The search process is Markovian and the probability of locating a customer in (t, t + dt) is proportional to the number of customers in the pool at time t. Various stationary probability distributions for this model are obtained in explicit forms well-suited for numerical computation.Under the assumption of exponential service times, corresponding results are obtained for the case where customers may escape from the pool.

A service model in which the server is required to search for customers

1984 ◽  
Vol 21 (01) ◽  
pp. 157-166 ◽  
Author(s):  
Marcel F. Neuts ◽  
M. F. Ramalhoto
Keyword(s):  
Poisson Process ◽  
Numerical Computation ◽  
Probability Distributions ◽  
General Distribution ◽  
Search Process ◽  
Single Server ◽  
Stationary Probability ◽  
Service Model ◽  
Service Times ◽  
Number Of Customers

Customers enter a pool according to a Poisson process and wait there to be found and processed by a single server. The service times of successive items are independent and have a common general distribution. Successive services are separated by seek phases during which the server searches for the next customer. The search process is Markovian and the probability of locating a customer in (t, t + dt) is proportional to the number of customers in the pool at time t. Various stationary probability distributions for this model are obtained in explicit forms well-suited for numerical computation. Under the assumption of exponential service times, corresponding results are obtained for the case where customers may escape from the pool.

Extremal properties of the FIFO discipline in queueing networks

1992 ◽  
Vol 29 (4) ◽  
pp. 967-978 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar
Keyword(s):  
Likelihood Ratio ◽  
Service Time ◽  
Queueing Networks ◽  
Service Rate ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queues ◽  
First Results ◽  
Service Times ◽  
Number Of Customers

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

scholarly journals Discrete- Time Queueing Model Geox/G/ꚙ with Bulk Arrival Rule

2020 ◽  
Vol 9 (3) ◽  
pp. 2585-2589
Keyword(s):  
Discrete Time ◽  
Geometric Distribution ◽  
Transient State ◽  
State Solution ◽  
General Distribution ◽  
Queueing Model ◽  
Service Times ◽  
Bulk Arrival ◽  
Number Of Customers

A discrete time queueing model is considered to estimate of the number of customers in the system. The arrivals, which are in groups of size X, inter-arrivals times and service times are distributed independent. The inter-arrivals fallows geometric distribution with parameter p and service times follows general distribution with parameter µ, we have derive the various transient state solution along with their moments and numerical illustrations in this paper.

scholarly journals Asymptotic Diffusion Analysis of Multi-Server Retrial Queue with Hyper-Exponential Service

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 531 ◽  
Author(s):  
Alexander Moiseev ◽  
Anatoly Nazarov ◽  
Svetlana Paul
Keyword(s):  
Numerical Analysis ◽  
Diffusion Process ◽  
Service Time ◽  
Probability Distributions ◽  
Retrial Queue ◽  
Stationary Probability ◽  
Diffusion Analysis ◽  
Number Of Customers ◽  
Multi Server

A multi-server retrial queue with a hyper-exponential service time is considered in this paper. The study is performed by the method of asymptotic diffusion analysis under the condition of long delay in orbit. On the basis of the constructed diffusion process, we obtain approximations of stationary probability distributions of the number of customers in orbit and the number of busy servers. Using simulations and numerical analysis, we estimate the accuracy and applicability area of the obtained approximations.

Basic analysis of a head-of-the-line priority queue with renewal type input

1975 ◽  
Vol 12 (2) ◽  
pp. 346-352
Author(s):  
R. Schassberger
Keyword(s):  
Distribution Function ◽  
Priority Queue ◽  
General Distribution ◽  
Single Server ◽  
Service Times

A single server is fed by a renewal stream of individual customers. These are of type k with probability πk, k = 1, …, N, and are all served individually. Upon completion of a service the server proceeds immediately with a customer of the lowest type (= highest priority) present, if any. Service times for type k are drawn from a general distribution function Bk(t) concentrated on (0, ∞).We lay the foundations for a broad analysis of the model.

scholarly journals The stationary local sojourn time in single server tandem queues with renewal input

1999 ◽  
Vol 12 (4) ◽  
pp. 417-428
Author(s):  
Pierre Le Gall
Keyword(s):  
Sojourn Time ◽  
Stationary Regime ◽  
General Distribution ◽  
Single Server ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Service Times

We start from an earlier paper evaluating the overall sojourn time to derive the local sojourn time in stationary regime, in a single server tandem queue of (m+1) stages with renewal input. The successive service times of a customer may or may not be mutually dependent, and are governed by a general distribution which may be different at each sage.

scholarly journals Retrial queues with recurrent demand option

1996 ◽  
Vol 9 (2) ◽  
pp. 221-228 ◽  
Author(s):  
K. Farahmand ◽  
N. H. Smith
Keyword(s):  
Poisson Process ◽  
Queueing System ◽  
Service System ◽  
Queueing Systems ◽  
Retrial Queues ◽  
General Distribution ◽  
Fixed Rate ◽  
Service Times ◽  
Request Service

The object of this paper is to analyze the model of a queueing system in which customers can call in only to request service: if the server is free, the customer enters service immediately. Otherwise, if the service system is occupied, the customer joins a source of unsatisfied customers called the orbit. On completion of each service the recipient of service has an option of leaving the system completely with probability 1−p or returning to the orbit with probability p. We consider two models characterized by the discipline governing the order of rerequests for service from the orbit. First, all the customers from the orbit apply at a fixed rate. Secondly, customers from the orbit are discouraged and reduce their rate of demand as more customers join the orbit. The arrival at and the demands from the orbit are both assumed to be according to the Poisson process. However, the service times for both primary customers and customers from the orbit are assumed to have a general distribution. We calculate several characteristic quantities of these queueing systems.

Extremal properties of the FIFO discipline in queueing networks

1992 ◽  
Vol 29 (04) ◽  
pp. 967-978 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar
Keyword(s):  
Likelihood Ratio ◽  
Service Time ◽  
Queueing Networks ◽  
Service Rate ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queues ◽  
First Results ◽  
Service Times ◽  
Number Of Customers

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons: (i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means. (ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means. We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR. Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

Optimal Server Allocation to Parallel Queues with Finite-Capacity Buffers

1996 ◽  
Vol 10 (2) ◽  
pp. 279-285 ◽  
Author(s):  
K. M. Wasserman ◽  
Nicholas Bambos
Keyword(s):  
Arbitrary Distribution ◽  
Single Server ◽  
Finite Capacity ◽  
Dynamic Allocation ◽  
Allocation Policy ◽  
Parallel Queues ◽  
Server Allocation ◽  
Arrival Processes ◽  
Service Times ◽  
Number Of Customers

In this paper, we study the problem of dynamic allocation of a single server to parallel queues with finite-capacity buffers. The arrival processes are mutually independent, equal rate Poisson processes, and the service times are independent and identically distributed random variables with an arbitrary distribution. We are interested in characterizing the allocation policy that stochastically minimizes the number of customers lost due to buffer overflows. Using a coupling argument, we establish the optimality of the Fewest-Empty-Spaces policy, which allocates the server to the queue with the fewest empty buffer spaces, within the class of nonpreemptive and nonidling policies. The result extends to the class of preemptive policies, if the service times are exponentially distributed. We also briefly discuss the allocation problem under more general statistical assumptions on the arrival processes.

scholarly journals The optimal control in batch arrival queue with server vacations, startup and breakdowns

2004 ◽  
Vol 14 (1) ◽  
pp. 41-55 ◽  
Author(s):  
Jau-Chuan Ke
Keyword(s):  
Optimal Control ◽  
Poisson Process ◽  
Cost Function ◽  
Minimum Cost ◽  
Compound Poisson Process ◽  
General Distribution ◽  
Server Vacations ◽  
Startup Time ◽  
Number Of Customers ◽  
System Characteristics

This paper studies the N policy M[x]/G/1 queue with server vacations; startup and breakdowns, where arrivals form a compound Poisson process and service times are generally distributed. The server is turned off and takes a vacation whenever the system is empty. If the number of customers waiting in the system at the instant of a vacation completion is less than N, the server will take another vacation. If the server returns from a vacation and finds at least N customers in the system, he is immediately turned on and requires a startup time before providing the service until the system is empty again. It is assumed that the server breaks down according to a Poisson process whose repair time has a general distribution. The system characteristics of such a model are analyzed and the total expected cost function per unit time is developed to determine the optimal threshold of N at a minimum cost.

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