scholarly journals Analysis of the Optimal Resource Allocation for a Tandem Queueing System

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Zaiming Liu ◽  
Wei Deng ◽  
Gang Chen
Keyword(s):  
Optimal Policy ◽  
Optimal Allocation ◽  
Queueing System ◽  
Service Rate ◽  
Allocation Policy ◽  
Long Run ◽  
Markov Decision ◽  
Upstream Station ◽  
Optimal Resource ◽  
Resource Cost

We study a controllable two-station tandem queueing system, where customers (jobs) must first be processed at upstream station and then the downstream station. A manager dynamically allocates the service resource to each station to adjust the service rate, leading to a tradeoff between the holding cost and resource cost. The goal of the manager is to find the optimal policy to minimize the long-run average costs. The problem is constructed as a Markov decision process (MDP). In this paper, we consider the model in which the resource cost and service rate functions are more general than linear. We derive the monotonicity of the optimal allocation policy by the quasiconvexity properties of the value function. Furthermore, we obtain the relationship between the two stations’ optimal policy and conditions under which the optimal policy is unique and has the bang-bang control property. Finally, we provide some numerical experiments to illustrate these results.

scholarly journals Approximations in Performance Analysis of a Controllable Queueing System with Heterogeneous Servers

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1803
Author(s):  
Dmitry Efrosinin ◽  
Natalia Stepanova ◽  
Janos Sztrik ◽  
Andreas Plank
Keyword(s):  
Optimal Policy ◽  
Optimal Allocation ◽  
Queueing System ◽  
Control Policy ◽  
Iteration Algorithm ◽  
Heuristic Solution ◽  
Allocation Policy ◽  
Markov Decision Model ◽  
The Mean ◽  
Number Of Customers

The paper studies a controllable multi-server heterogeneous queueing system where servers operate at different service rates without preemption, i.e., the service times are uninterrupted. The optimal control policy allocates the customers between the servers in such a way that the mean number of customers in the system reaches its minimal value. The Markov decision model and the policy-iteration algorithm are used to calculate the optimal allocation policy and corresponding mean performance characteristics. The optimal policy, when neglecting the weak influence of slow servers, is of threshold type defined as a sequence of threshold levels which specifies the queue lengths for the usage of any slower server. To avoid time-consuming calculations for systems with a large number of servers, we focus here on a heuristic evaluation of the optimal thresholds and compare this solution with the real values. We develop also the simple lower and upper bound methods based on approximation by an equivalent heterogeneous queueing system with a preemption to measure the mean number of customers in the system operating under the optimal policy. Finally, the simulation technique is used to provide sensitivity analysis of the heuristic solution to changes in the form of inter-arrival and service time distributions.

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.

Tractable Equilibria in Sponsored Search with Endogenous Budgets

2020 ◽  
Author(s):  
Dragos Florin Ciocan ◽  
Krishnamurthy Iyer
Keyword(s):  
Large Scale ◽  
Optimal Allocation ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Sponsored Search ◽  
Allocation Policy ◽  
Long Run ◽  
Linear Optimization Problems ◽  
Search Market ◽  
The Individual

Given the scale of the sponsored search market, it is practically important yet technically difficult to understand the interplay between bidders and the ad network and its effect on the long-run state of the market. Although typical equilibrium models account for bidders strategizing over the individual bids they submit to the auctions, they ignore that bidders also strategically set their campaign budgets. In “Tractable Equilibria in Sponsored Search with Endogenous Budgets,” F. Ciocan and K. Iyer ask how this additional strategic layer affects market operation and prove that endogenizing budgets surprisingly yields simple and interpretable equilibria. Namely, these equilibria generate quasi-truthful bidding strategies guaranteeing bidders an ROI exceeding their cost per dollar of committed budget. Additionally, the ad network’s optimal allocation policy becomes greedy with high probability. Thus, in this equilibrium, the ad network need not solve computationally challenging, large-scale linear optimization problems typically required under exogenous budgets.

Optimal service-rate control of M/G/1 queueing systems using phase methods

1983 ◽  
Vol 15 (3) ◽  
pp. 616-637 ◽  
Author(s):  
Kyung Y. Jo ◽  
Shaler Stidham
Keyword(s):  
Partial Information ◽  
Queueing Systems ◽  
Infinite Horizon ◽  
Service Rate ◽  
Erlang Distribution ◽  
Work Requirement ◽  
Average Return ◽  
Long Run ◽  
Markov Decision ◽  
Method Of Phases

A new approach to the optimal control of the service rate in M/G/1 queues is introduced using the method of phases. Each customer's work requirement is approximated by a random number of exponential phases with (possibly) different parameters (a generalized hyper-Erlang distribution). Using a semi-Markov decision-process formulation, we establish monotonicity properties of optimal policies for the finite-horizon problem, by induction on the horizon length. The analysis is then extended to the discounted infinite-horizon and the long-run average-return problems. In contrast to the models in previous papers, our model is appropriate for situations where the system controller has partial information about the work requirement of a customer, specifically the number of phases (tasks) to be performed. Because it requires a multidimensional state description, the analysis of the phase-type control model may be viewed as a first step toward the solution of control models for networks of queues.

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.

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.

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

2001 ◽  
Vol 38 (02) ◽  
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.

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.

On the Optimality of Trunk Reservation in Overflow Processes

1991 ◽  
Vol 5 (3) ◽  
pp. 369-390 ◽  
Author(s):  
Viên Nguyen
Keyword(s):  
Poisson Process ◽  
Optimal Policy ◽  
Waiting Room ◽  
Long Run ◽  
Markov Decision ◽  
Customer Type ◽  
Reservation Policy ◽  
Optimal Dynamic ◽  
Trunk Reservation ◽  
Overflow Process

This paper discusses an optimal dynamic policy for a queueing system with M servers, no waiting room, and two types of customers. Customer types differ with respect to the reward that is paid on commencement of service, but service times are exponentially distributed with the same mean for both types of customers. The arrival stream of one customer type is generated by a Poisson process, and the other customer type arrives according to the overflow process of an M/M/m/m queue. The objective is to determine a policy for admitting customers to maximize the expected long-run average reward.By posing the problem in the framework of Markov decision processes and exploiting properties of submodular functions, the optimal policy is shown to be a “generalized trunk reservation policy”; in other words, the optimal policy accepts higher-paying customers whenever possible and accepts lower-paying customers only if fewer than c1 servers are busy, where i is the number of busy servers in the overflow queue. Computational issues are also discussed. More specifically, approximations of the overflow process by an interrupted Poisson process and a Poisson process are investigated.

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.

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