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.

Dynamic pricing in a two-class queueing system with arrival and service rate control

2020 ◽  
Author(s):  
Shuangfeng Ma ◽  
Wei Guo
Keyword(s):  
Rate Control ◽  
Dynamic Pricing ◽  
Queueing System ◽  
Recursive Algorithm ◽  
Optimization Theory ◽  
Service Rate ◽  
Long Run ◽  
Holding Cost ◽  
Service Rates ◽  
Number Of Customers

Abstract Dynamic pricing in a two-class queueing system with adjustable arrival and service rates is considered in this paper. We initially take the adjustable rates into account to maximize the long-run average social welfare and further establish matched dynamic prices to lead two distinct types of customers’ behavior. For the rate-setting problems, we apply the sensitivity-based optimization theory and an iterative algorithm to investigate the two types of customers’ optimal arrival and service rates. Next, we apply the results obtained from rate-setting problems to acquire the expected delay time by recursive algorithm and demonstrate the optimal prices formulas for multiple customers explicitly. Finally, we carry out some numerical experiments to illustrate our consequence and the performance between two kinds of customers with different level of holding cost. It appears that under low holding cost, the optimal prices for two kinds of customers are monotonically increasing in the number of customers regardless of classes, but under high holding cost, the optimal prices for the customers who have low waiting cost may drop when the number of the other class rises.

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.

Bias optimal admission control policies for a multiclass nonstationary queueing system

2002 ◽  
Vol 39 (01) ◽  
pp. 20-37 ◽  
Author(s):  
Mark E. Lewis ◽  
Hayriye Ayhan ◽  
Robert D. Foley
Keyword(s):  
Optimal Policy ◽  
Queueing System ◽  
Sufficient Conditions ◽  
System Capacity ◽  
Average Reward ◽  
Finite Capacity ◽  
Long Run ◽  
Optimal Policies ◽  
Service Rates ◽  
Long Run Average Reward

We consider a finite-capacity queueing system where arriving customers offer rewards which are paid upon acceptance into the system. The gatekeeper, whose objective is to ‘maximize’ rewards, decides if the reward offered is sufficient to accept or reject the arriving customer. Suppose the arrival rates, service rates, and system capacity are changing over time in a known manner. We show that all bias optimal (a refinement of long-run average reward optimal) policies are of threshold form. Furthermore, we give sufficient conditions for the bias optimal policy to be monotonic in time. We show, via a counterexample, that if these conditions are violated, the optimal policy may not be monotonic in time or of threshold form.

scholarly journals Markovian Queueing System with Discouraged Arrivals and Self-Regulatory Servers

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
K. V. Abdul Rasheed ◽  
M. Manoharan
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Expected Number ◽  
Multiple Servers ◽  
Special Cases ◽  
Service Rates ◽  
Expected Waiting Time ◽  
Number Of Customers ◽  
Steady State Probabilities

We consider discouraged arrival of Markovian queueing systems whose service speed is regulated according to the number of customers in the system. We will reduce the congestion in two ways. First we attempt to reduce the congestion by discouraging the arrivals of customers from joining the queue. Secondly we reduce the congestion by introducing the concept of service switches. First we consider a model in which multiple servers have three service ratesμ1,μ2, andμ(μ1≤μ2<μ), say, slow, medium, and fast rates, respectively. If the number of customers in the system exceeds a particular pointK1orK2, the server switches to the medium or fast rate, respectively. For this adaptive queueing system the steady state probabilities are derived and some performance measures such as expected number in the system/queue and expected waiting time in the system/queue are obtained. Multiple server discouraged arrival model having one service switch and single server discouraged arrival model having one and two service switches are obtained as special cases. A Matlab program of the model is presented and numerical illustrations are given.

Confidence Intervals for Performance Measures of M/M/1 Queue with Constant Retrial Policy

2015 ◽  
Vol 32 (06) ◽  
pp. 1550046
Author(s):  
Dmitry Efrosinin ◽  
Anastasia Winkler ◽  
Pinzger Martin
Keyword(s):  
Performance Measures ◽  
Queueing System ◽  
Retrial Queue ◽  
Threshold Level ◽  
Stationary Regime ◽  
Parametric Estimation ◽  
Threshold Policy ◽  
System Parameters ◽  
Number Of Customers ◽  
The Given

We consider the problem of estimation and confidence interval construction of a Markovian controllable queueing system with unreliable server and constant retrial policy. For the fully observable system the standard parametric estimation technique is used. The arrived customer finding a free server either gets service immediately or joins a retrial queue. The customer at the head of the retrial queue is allowed to retry for service. When the server is busy, it is subject to breakdowns. In a failed state the server can be repaired with respect to the threshold policy: the repair starts when the number of customers in the system reaches a fixed threshold level. To obtain the estimates for the system parameters, performance measures and optimal threshold level we analyze the system in a stationary regime. The performance measures including average cost function for the given cost structure are presented in a closed matrix form.

ANALYSIS OF A BULK QUEUE WITH FAST AND SLOW SERVICE RATES AND MULTIPLE VACATIONS

2005 ◽  
Vol 22 (02) ◽  
pp. 239-260 ◽  
Author(s):  
R. ARUMUGANATHAN ◽  
K. S. RAMASWAMI
Keyword(s):  
Performance Measures ◽  
Queue Length ◽  
Queueing System ◽  
Cost Model ◽  
Probability Generating Function ◽  
Service Rate ◽  
Multiple Vacations ◽  
Bulk Queue ◽  
Service Rates ◽  
Number Of Customers

We analyze a Mx/G(a,b)/1 queueing system with fast and slow service rates and multiple vacations. The server does the service with a faster rate or a slower rate based on the queue length. At a service completion epoch (or) at a vacation completion epoch if the number of customers waiting in the queue is greater than or equal to N (N > b), then the service is rendered at a faster rate, otherwise with a slower service rate. After finishing a service, if the queue length is less than 'a' the server leaves for a vacation of random length. When he returns from the vacation, if the queue length is still less than 'a' he leaves for another vacation and so on until he finally finds atleast 'a' customers waiting for service. After a service (or) a vacation, if the server finds atleast 'a' customers waiting for service say ξ, then he serves a batch of min (ξ, b) customers, where b ≥ a. We derive the probability generating function of the queue size at an arbitrary time. Various performance measures are obtained. A cost model is discussed with a numerical solution.

Bias optimal admission control policies for a multiclass nonstationary queueing system

2002 ◽  
Vol 39 (1) ◽  
pp. 20-37 ◽  
Author(s):  
Mark E. Lewis ◽  
Hayriye Ayhan ◽  
Robert D. Foley
Keyword(s):  
Optimal Policy ◽  
Queueing System ◽  
Sufficient Conditions ◽  
System Capacity ◽  
Average Reward ◽  
Finite Capacity ◽  
Long Run ◽  
Optimal Policies ◽  
Service Rates ◽  
Long Run Average Reward

We consider a finite-capacity queueing system where arriving customers offer rewards which are paid upon acceptance into the system. The gatekeeper, whose objective is to ‘maximize’ rewards, decides if the reward offered is sufficient to accept or reject the arriving customer. Suppose the arrival rates, service rates, and system capacity are changing over time in a known manner. We show that all bias optimal (a refinement of long-run average reward optimal) policies are of threshold form. Furthermore, we give sufficient conditions for the bias optimal policy to be monotonic in time. We show, via a counterexample, that if these conditions are violated, the optimal policy may not be monotonic in time or of threshold form.

Time-dependent solution and optimal control of a bulk service queue

1997 ◽  
Vol 34 (01) ◽  
pp. 258-266
Author(s):  
Shokri Z. Selim
Keyword(s):  
Queueing System ◽  
Maximum Size ◽  
Time Dependent ◽  
Optimal Service ◽  
Bulk Service ◽  
Service Rates ◽  
Finite Time Horizon ◽  
Number Of Customers ◽  
Time Dependent Solution ◽  
Dependent Probability

We consider the queueing system denoted by M/MN /1/N where customers are served in batches of maximum size N. The model is motivated by a traffic application. The time-dependent probability distribution for the number of customers in the system is obtained in closed form. The solution is used to predict the optimal service rates during a finite time horizon.

Average optimal policies in a controlled queueing system with dual admission control

2001 ◽  
Vol 38 (2) ◽  
pp. 369-385 ◽  
Author(s):  
Mark E. Lewis
Keyword(s):  
Admission Control ◽  
Manufacturing Systems ◽  
Queueing System ◽  
Service Rate ◽  
Average Reward ◽  
Long Run ◽  
Switching Curve ◽  
Multiple Tasks ◽  
Optimal Policies ◽  
Number Of Customers

We consider a controlled M/M/1 queueing system where customers may be subject to two potential rejections. The first occurs upon arrival and is dependent on the number of customers in the queue and the service rate of the customer currently in service. The second, which may or may not occur, occurs immediately prior to the customer receiving service. That is, after each service completion the customer in the front of the queue is assessed and the service rate of that customer is revealed. If the second decision-maker recommends rejection, the customer is denied service with a fixed probability. We show the existence of long-run average optimal monotone switching-curve policies. Further, we show that the average reward is increasing in the probability that the second decision-maker's recommendation of rejection is honored. Applications include call centers with delayed classifications and manufacturing systems when the server is responsible for multiple tasks.

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