scholarly journals Discrete Time Multi-servers Loss Systems and Stochastic Approximation with Delayed Groups of Customers

2021 ◽  
Vol 19 (6) ◽  
pp. 575-583
Author(s):  
Rasha Atwa ◽  
Rasha Abd- El - Wahab ◽  
Ola Barakat
Keyword(s):  
Applied Sciences ◽  
Stochastic Approximation ◽  
Service Time ◽  
Random Variable ◽  
Fixed Time ◽  
Approximation Procedure ◽  
Stochastic Approximation Procedure ◽  
Loss Systems ◽  
Service Times ◽  
Number Of Customers

The stochastic approximation procedure with delayed groups of delayed customers is investigated. The Robbins-Monro stochastic approximation procedure is adjusted to be usable in the presence of delayed groups of delayed customers. Two loss systems are introduced to get an accurate description of the proposed procedure. Each customer comes after fixed time-intervals with the stage of the following customer is accurate according to the outcome of the preceding one, where the serving time of a customer is assumed to be discrete random variable. Some applications of the procedure are given where the analysis of their results is obtained. The analysis shows that efficiencies of the procedure can be increased by minimizing the number of customers of a group irrespective of their service times that may take maximum values. Efficiencies depend on the maximum service time of the customer and on the number of customers of the group. The most important result is that efficiencies of the procedure are increased by increasing the service time distributions as well as service times of customers .This new situation can be applied to increase the number of served customers where the number of served groups will also be increased. The results obtained seem to be acceptable. In general, our proposal can be utilized to other stochastic approximation procedures to increase the production in many fields such as medicine, computer sciences, industry, and applied sciences.

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 On the almost sure convergence of a general stochastic approximation procedure

1986 ◽  
Vol 34 (3) ◽  
pp. 335-342 ◽  
Author(s):  
S. N. Evans ◽  
N. C. Weber
Keyword(s):  
Regression Model ◽  
Stochastic Approximation ◽  
Iterative Procedure ◽  
Almost Sure Convergence ◽  
Approximation Procedure ◽  
Stochastic Approximation Procedure ◽  
General Stochastic

A set of conditions for the almost sure convergence of a stochastic iterative procedure is given. The conditions are framed in terms of the behaviour of the random adjustment made the n-th step rather than in terms of some underlying regression model.

scholarly journals On the optimal assignment of customers to parallel servers

1978 ◽  
Vol 15 (02) ◽  
pp. 406-413 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Stochastic Process ◽  
Service Time ◽  
Hazard Rate ◽  
Random Variable ◽  
Queuing System ◽  
Optimal Assignment ◽  
Parallel Servers ◽  
Shortest Queue ◽  
Number Of Customers

We consider a queuing system with several identical servers, each with its own queue. Identical customers arrive according to some stochastic process and as each customer arrives it must be assigned to some server's queue. No jockeying amongst the queues is allowed. We are interested in assigning the arriving customers so as to maximize the number of customers which complete their service by a certain time. If each customer's service time is a random variable with a non-decreasing hazard rate then the strategy which does this is one which assigns each arrival to the shortest queue.

Computational analysis of single-server bulk-service queues, M/GY/ 1

1989 ◽  
Vol 21 (1) ◽  
pp. 207-225 ◽  
Author(s):  
G. Brière ◽  
M. L. Chaudhry
Keyword(s):  
Service Time ◽  
Flexible Manufacturing ◽  
Time Distribution ◽  
Random Variable ◽  
Single Server ◽  
Service Time Distribution ◽  
Computationally Efficient ◽  
Uniform Distributions ◽  
Bulk Service ◽  
Number Of Customers

Algorithms are proposed for the numerical inversion of the analytical solutions obtained through classical transform methods. We compute steady-state probabilities and moments of the number of customers in the system (or in the queue) at three different epochs—postdeparture, random, and prearrival—for models of the type M/GY/1, where the capacity of the single server is a random variable. This implies first finding roots of the characteristic equation, which is detailed in an appendix for a general service time distribution. Numerical results, given a service time distribution, are illustrated through graphs and tables for cases covered in this study: deterministic, Erlang, hyperexponential, and uniform distributions. In all cases, the proposed method is computationally efficient and accurate, even for high values of the queueing parameters. The procedure is adaptable to other models in queueing theory (especially bulk queues), to problems in inventory control, transportation, flexible manufacturing process, etc. Exact results that can be obtained from the algorithms presented here will be found useful to test inequalities, bounds, or approximations.

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