scholarly journals Algorithmic Analysis of Finite-Source Multi-Server Heterogeneous Queueing Systems

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2624
Author(s):  
Dmitry Efrosinin ◽  
Natalia Stepanova ◽  
Janos Sztrik
Keyword(s):  
Optimal Allocation ◽  
Queueing Systems ◽  
Mean Value ◽  
Threshold Policy ◽  
Heterogeneous Model ◽  
Finite Source ◽  
Long Run ◽  
The Matrix ◽  
Number Of Customers ◽  
Main Model

The paper deals with a finite-source queueing system serving one class of customers and consisting of heterogeneous servers with unequal service intensities and of one common queue. The main model has a non-preemptive service when the customer can not change the server during its service time. The optimal allocation problem is formulated as a Markov-decision one. We show numerically that the optimal policy which minimizes the long-run average number of customers in the system has a threshold structure. We derive the matrix expressions for performance measures of the system and compare the main model with alternative simplified queuing systems which are analysed for the arbitrary number of servers. We observe that the preemptive heterogeneous model operating under a threshold policy is a good approximation for the main model by calculating the mean number of customers in the system. Moreover, using the preemptive and non-preemptive queueing models with the faster server first policy the lower and upper bounds are calculated for this mean value.

OPTIMAL CONTROL OF A TWO-SERVER QUEUEING SYSTEM WITH FAILURES

2014 ◽  
Vol 28 (4) ◽  
pp. 489-527 ◽  
Author(s):  
Erhun Özkan ◽  
Jeffrey P. Kharoufeh
Keyword(s):  
Process Model ◽  
Queueing System ◽  
Threshold Policy ◽  
Long Run ◽  
Reliability Characteristics ◽  
Markov Decision ◽  
Service Rates ◽  
Number Of Customers ◽  
Random Failures ◽  
Main Model

We consider the problem of controlling a two-server Markovian queueing system with heterogeneous servers. The servers are differentiated by their service rates and reliability attributes (i.e., the slower server is perfectly reliable, whereas the faster server is subject to random failures). The aim is to dynamically route customers at arrival, service completion, server failure, and server repair epochs to minimize the long-run average number of customers in the system. Using a Markov decision process model, we prove that it is always optimal to route customers to the faster server when it is available, irrespective of its failure and repair rates, if the system is stable. For the slower server, there exists an optimal threshold policy that depends on the queue length and the state of the faster server. Additionally, we analyze a variant of the main model in which there are multiple unreliable servers with identical service rates, but distinct reliability characteristics. For that case it is always optimal to route customers to idle servers, and the optimal policy is insensitive to the servers’ reliability characteristics.

A comparison of the stationary distributions of GI/M/c/n and GI/M/c

1998 ◽  
Vol 35 (02) ◽  
pp. 510-515
Author(s):  
F. Simonot
Keyword(s):  
Unique Solution ◽  
Convergence Rates ◽  
Queueing Systems ◽  
Mean Value ◽  
Interarrival Time ◽  
Traffic Intensity ◽  
Stationary Distributions ◽  
Number Of Customers

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/c/n and GI/M/c queueing systems as n tends to infinity. We show that earlier results established for GI/M/1/n and GI/M/1 remain true. Namely, it is proved that if the interarrival time c.d.f. H is non lattice with mean value λ−1 and if the traffic intensity is strictly less than one, then the convergence rates in l 1 norm of the arrival and time stationary distributions of GI/M/c/n to the corresponding stationary distributions of GI/M/c are geometric and are characterized by ω, the unique solution in (0,1) of the equation z = ∫∞ 0 exp{-μc(1-z)t}dH(t).

scholarly journals Optimal Open-Loop Routing and Threshold-Based Allocation in TWO Parallel QUEUEING Systems with Heterogeneous Servers

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2766
Author(s):  
Dmitry Efrosinin ◽  
Natalia Stepanova
Keyword(s):  
Optimal Allocation ◽  
Queueing Systems ◽  
Open Loop ◽  
Arrival Process ◽  
Optimal Routing ◽  
Threshold Policy ◽  
Loop Control ◽  
Allocation Policy ◽  
Routing Policy ◽  
Markov Modulated

In this paper, we study the problem of optimal routing for the pair of two-server heterogeneous queues operating in parallel and subsequent optimal allocation of customers between the servers in each queue. Heterogeneity implies different servers in terms of speed of service. An open-loop control assumes the static resource allocation when a router has no information about the state of the system. We discuss here the algorithm to calculate the optimal routing policy based on specially constructed Markov-modulated Poisson processes. As an alternative static policy, we consider an optimal Bernoulli splitting which prescribes the optimal allocation probabilities. Then, we show that the optimal allocation policy between the servers within each queue is of threshold type with threshold levels depending on the queue length and phase of an arrival process. This dependence can be neglected by using a heuristic threshold policy. A number of illustrative examples show interesting properties of the systems operating under the introduced policies and their performance characteristics.

Convergence rate for the distributions of GI/M/1/n and M/GI/1/n as n tends to infinity

1997 ◽  
Vol 34 (4) ◽  
pp. 1049-1060 ◽  
Author(s):  
F. Simonot
Keyword(s):  
Convergence Rate ◽  
Unique Solution ◽  
Queueing Systems ◽  
Mean Value ◽  
Traffic Intensity ◽  
Stationary Distributions ◽  
Number Of Customers

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/1/n and GI/M/1 queueing systems. We show that, if the inter-arrival c.d.f. H is non-lattice with mean value λ –1, and if the traffic intensity ρ = λμ –1 is strictly less than one, then the convergence rate of the stationary distributions of GI/M/1/n to the corresponding stationary distributions of GI/M/1 is geometric. More-over, the convergence rate can be characterized by the number ω, the unique solution in (0, 1) of the equation . A similar result is established for the M/GI/1/n and M/GI/1 queueing systems.

A comparison of the stationary distributions of GI/M/c/n and GI/M/c

1998 ◽  
Vol 35 (2) ◽  
pp. 510-515 ◽  
Author(s):  
F. Simonot
Keyword(s):  
Unique Solution ◽  
Convergence Rates ◽  
Queueing Systems ◽  
Mean Value ◽  
Interarrival Time ◽  
Traffic Intensity ◽  
Stationary Distributions ◽  
Number Of Customers

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/c/n and GI/M/c queueing systems as n tends to infinity. We show that earlier results established for GI/M/1/n and GI/M/1 remain true. Namely, it is proved that if the interarrival time c.d.f. H is non lattice with mean value λ−1 and if the traffic intensity is strictly less than one, then the convergence rates in l1norm of the arrival and time stationary distributions of GI/M/c/n to the corresponding stationary distributions of GI/M/c are geometric and are characterized by ω, the unique solution in (0,1) of the equation z = ∫∞0 exp{-μc(1-z)t}dH(t).

Convergence rate for the distributions of GI/M/1/n and M/GI/1/n as n tends to infinity

1997 ◽  
Vol 34 (04) ◽  
pp. 1049-1060 ◽  
Author(s):  
F. Simonot
Keyword(s):  
Convergence Rate ◽  
Unique Solution ◽  
Queueing Systems ◽  
Mean Value ◽  
Traffic Intensity ◽  
Stationary Distributions ◽  
Number Of Customers

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/1/n and GI/M/1 queueing systems. We show that, if the inter-arrival c.d.f. H is non-lattice with mean value λ – 1 , and if the traffic intensity ρ = λμ – 1 is strictly less than one, then the convergence rate of the stationary distributions of GI/M/1/n to the corresponding stationary distributions of GI/M/1 is geometric. More-over, the convergence rate can be characterized by the number ω, the unique solution in (0, 1) of the equation . A similar result is established for the M/GI/1/n and M/GI/1 queueing systems.

Heterogeneous model of schistosomiasis transmission and long-term control: the combined influence of spatial variation and age-dependent factors on optimal allocation of drug therapy

Parasitology ◽  
2004 ◽  
Vol 130 (1) ◽  
pp. 49-65 ◽  
Author(s):  
D. GURARIE ◽  
C. H. KING
Keyword(s):  
Large Scale ◽  
Optimal Allocation ◽  
Field Studies ◽  
Human Populations ◽  
Infection Prevalence ◽  
Snail Population ◽  
Extended Model ◽  
Heterogeneous Model ◽  
Age Dependent

Prior field studies and modelling analyses have individually highlighted the importance of age-specific and spatial heterogeneities on the risk for schistosomiasis in human populations. As long-term, large-scale drug treatment programs for schistosomiasis are initiated in subSaharan Africa and elsewhere, optimal strategies for timing and distribution of therapy have yet to be fully defined on the working, district-level scale, where strong heterogeneities are often observed among sublocations. Based on transmission estimates from recent field studies, we develop an extended model of heterogeneous schistosome transmission for distributed human and snail population clusters and age-dependent behaviour, based on a ‘mean worm burden+snail infection prevalence’ formulation. We analyse its equilibria and basic reproduction patterns and their dependence on the underlying transmission parameters. Our model allows the exploration of chemotherapy-based control strategies targeted at high-risk behavioural groups and localities, and the approach to an optimal design in terms of cost. Efficacy of the approach is demonstrated for a model environment having linked, but spatially-distributed, populations and transmission sites.

SINGLE-MACHINE MULTIPLE-RECIPE PREDICTIVE MAINTENANCE

2013 ◽  
Vol 27 (2) ◽  
pp. 209-235 ◽  
Author(s):  
Yiwei Cai ◽  
John J. Hasenbein ◽  
Erhan Kutanoglu ◽  
Melody Liao
Keyword(s):  
Optimal Policy ◽  
Predictive Maintenance ◽  
Production Costs ◽  
Threshold Policy ◽  
Discrete Time Markov Chain ◽  
System A ◽  
Long Run ◽  
Monotonicity Result ◽  
Maintenance Policies ◽  
Discounted Costs

This paper studies a multiple-recipe predictive maintenance problem with M/G/1 queueing effects. The server degrades according to a discrete-time Markov chain and we assume that the controller knows both the machine status and the current number of jobs in the system. The controller's objective is to minimize total discounted costs or long-run average costs which include preventative and corrective maintenance costs, holdings costs, and possibly production costs. An optimal policy determines both when to perform maintenance and which type of job to process. Since the policy takes into account the machine's degradation status, such control decisions are known as predictive maintenance policies. In the single-recipe case, we prove that the optimal policy is monotone in the machine status, but not in the number of jobs in the system. A similar monotonicity result holds in the two-recipe case. Finally, we provide computational results indicating that significant savings can be realized when implementing a predictive maintenance policies instead of a traditional job-based threshold policy for preventive maintenances.

The distribution of the maximum number of customers present simultaneously during a busy period for the queueing systems M/G/1 and G/M/1

1967 ◽  
Vol 4 (01) ◽  
pp. 162-179 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Busy Period ◽  
Transition Probabilities ◽  
Queueing Systems ◽  
Entrance Time ◽  
Distribution Of The Maximum ◽  
Number Of Customers

The distribution of the maximum number of customers simultaneously present during a busy period is studied for the queueing systems M/G/1 and G/M/1. These distributions are obtained by using taboo probabilities. Some new relations for transition probabilities and entrance time distributions are derived.

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