scholarly journals OPTIMALITY OF TRUNK RESERVATION FOR AN M/M/K/N QUEUE WITH SEVERAL CUSTOMER TYPES AND HOLDING COSTS

2011 ◽  
Vol 25 (4) ◽  
pp. 537-560 ◽  
Author(s):  
Eugene A. Feinberg ◽  
Fenghsu Yang
Keyword(s):  
Optimal Control ◽  
Control Level ◽  
Reward Structure ◽  
Cost Rate ◽  
Holding Costs ◽  
Optimal Policies ◽  
Customer Type ◽  
Holding Cost ◽  
Trunk Reservation ◽  
Average Rewards

In this article we study optimal admission to an M/M/k/N queue with several customer types. The reward structure consists of revenues collected from admitted customers and holding costs, both of which depend on customer types. This article studies average rewards per unit time and describes the structures of stationary optimal, canonical, bias optimal, and Blackwell optimal policies. Similar to the case without holding costs, bias optimal and Blackwell optimal policies are unique, coincide, and have a trunk reservation form with the largest optimal control level for each customer type. Problems with one holding cost rate have been studied previously in the literature.

scholarly journals Optimality of randomized trunk reservation for a problem with a single constraint

2006 ◽  
Vol 38 (01) ◽  
pp. 199-220 ◽  
Author(s):  
X. Fan-Orzechowski ◽  
E. A. Feinberg
Keyword(s):  
Optimal Policy ◽  
Blocking Probability ◽  
Finite Capacity ◽  
Single Constraint ◽  
Customer Type ◽  
Trunk Reservation ◽  
Lagrangian Optimization ◽  
Finite Capacity Queue ◽  
Average Rewards ◽  
Time Subject

We study an optimal admission of arriving customers to a Markovian finite-capacity queue, e.g. an M/M/c/Nqueue, with several customer types. The system managers are paid for serving customers and penalized for rejecting them. The rewards and penalties depend on customer type. The goal is to maximize the average rewards per unit time subject to the constraint on the average penalties per unit time. We provide a solution to this problem based on Lagrangian optimization. For a feasible problem, we show the existence of a randomized trunk reservation optimal policy with the acceptance thresholds for different customer types ordered according to a linear combination of the service rewards and rejection costs. In addition, we prove that any 1-randomized stationary optimal policy has this structure. In particular, we establish the structure of an optimal policy that maximizes the average rewards per unit time subject to the constraint on the blocking probability of either one of the customer types or a group of customer types pooled together.

scholarly journals Optimality of randomized trunk reservation for a problem with a single constraint

2006 ◽  
Vol 38 (1) ◽  
pp. 199-220 ◽  
Author(s):  
X. Fan-Orzechowski ◽  
E. A. Feinberg
Keyword(s):  
Optimal Policy ◽  
Blocking Probability ◽  
Finite Capacity ◽  
Single Constraint ◽  
Customer Type ◽  
Trunk Reservation ◽  
Lagrangian Optimization ◽  
Finite Capacity Queue ◽  
Average Rewards ◽  
Time Subject

We study an optimal admission of arriving customers to a Markovian finite-capacity queue, e.g. an M/M/c/N queue, with several customer types. The system managers are paid for serving customers and penalized for rejecting them. The rewards and penalties depend on customer type. The goal is to maximize the average rewards per unit time subject to the constraint on the average penalties per unit time. We provide a solution to this problem based on Lagrangian optimization. For a feasible problem, we show the existence of a randomized trunk reservation optimal policy with the acceptance thresholds for different customer types ordered according to a linear combination of the service rewards and rejection costs. In addition, we prove that any 1-randomized stationary optimal policy has this structure. In particular, we establish the structure of an optimal policy that maximizes the average rewards per unit time subject to the constraint on the blocking probability of either one of the customer types or a group of customer types pooled together.

Optimal control of the service rate in an M/G/1 queueing system

1978 ◽  
Vol 10 (3) ◽  
pp. 682-701 ◽  
Author(s):  
Bharat T. Doshi
Keyword(s):  
Optimal Control ◽  
Average Cost ◽  
Queueing System ◽  
Sufficient Conditions ◽  
Service Rate ◽  
Service Cost ◽  
Stationary Policy ◽  
Discounted Cost ◽  
Cost Rate ◽  
Holding Cost

We consider an M/G/1 queue in which the service rate is subject to control. The control is exercised continuously and is based on the observations of the residual workload process. For both the discounted cost and the average cost criteria we obtain conditions which are sufficient for a stationary policy to be optimal. When the service cost rate and the holding cost rates are non-decreasing and convex it is shown that these sufficient conditions are satisfied by a monotonic policy, thus showing its optimality.

Optimal control of the service rate in an M/G/1 queueing system

1978 ◽  
Vol 10 (03) ◽  
pp. 682-701 ◽  
Author(s):  
Bharat T. Doshi
Keyword(s):  
Optimal Control ◽  
Average Cost ◽  
Queueing System ◽  
Sufficient Conditions ◽  
Service Rate ◽  
Service Cost ◽  
Stationary Policy ◽  
Discounted Cost ◽  
Cost Rate ◽  
Holding Cost

We consider an M/G/1 queue in which the service rate is subject to control. The control is exercised continuously and is based on the observations of the residual workload process. For both the discounted cost and the average cost criteria we obtain conditions which are sufficient for a stationary policy to be optimal. When the service cost rate and the holding cost rates are non-decreasing and convex it is shown that these sufficient conditions are satisfied by a monotonic policy, thus showing its optimality.

OPTIMAL CONTROL OF FLEXIBLE SERVERS IN TWO TANDEM QUEUES WITH OPERATING COSTS

2007 ◽  
Vol 22 (1) ◽  
pp. 107-131 ◽  
Author(s):  
Dimitrios G. Pandelis
Keyword(s):  
Optimal Control ◽  
Optimal Policy ◽  
Optimal Allocation ◽  
Operating Costs ◽  
Queuing Systems ◽  
Tandem Queues ◽  
Holding Costs ◽  
Flexible Servers ◽  
Tandem Queuing ◽  
Dedicated Servers

We consider two-stage tandem queuing systems with dedicated servers in each station and flexible servers that can serve in both stations. We assume exponential service times, linear holding costs, and operating costs incurred by the servers at rates proportional to their speeds. Under conditions that ensure the optimality of nonidling policies, we show that the optimal allocation of flexible servers is determined by a transition-monotone policy. Moreover, we present conditions under which the optimal policy can be explicitly determined.

Optimal control of random walks, birth and death processes, and queues

1981 ◽  
Vol 13 (01) ◽  
pp. 61-83 ◽  
Author(s):  
Richard Serfozo
Keyword(s):  
Optimal Control ◽  
Random Walks ◽  
Queue Length ◽  
Transition Probabilities ◽  
Transition Rates ◽  
Average Reward ◽  
Birth And Death Processes ◽  
Optimal Policies ◽  
One Step ◽  
Increasing Functions

This is a study of simple random walks, birth and death processes, and M/M/s queues that have transition probabilities and rates that are sequentially controlled at jump times of the processes. Each control action yields a one-step reward depending on the chosen probabilities or transition rates and the state of the process. The aim is to find control policies that maximize the total discounted or average reward. Conditions are given for these processes to have certain natural monotone optimal policies. Under such a policy for the M/M/s queue, for example, the service and arrival rates are non-decreasing and non-increasing functions, respectively, of the queue length. Properties of these policies and a linear program for computing them are also discussed.

Comparison of sequential and continuous inspection strategies for deteriorating systems

1994 ◽  
Vol 26 (2) ◽  
pp. 423-435 ◽  
Author(s):  
C. Teresa Lam ◽  
R. H. Yeh
Keyword(s):  
Continuous Time ◽  
Iterative Algorithms ◽  
Optimal Cost ◽  
Cost Rate ◽  
State Dependent ◽  
Long Run ◽  
Inspection Strategy ◽  
Optimal Policies ◽  
Finite State ◽  
Deteriorating Systems

This paper investigates inspection strategies for a finite-state continuous-time Markovian deteriorating system. Two inspection strategies are considered: sequential inspection and continuous inspection. Unlike many previous efforts, the inspection times for the sequential inspection strategy are assumed to be non-negligible. The replacement times and costs for both strategies are non-negligible and state dependent. Our objective here is to minimize the expected long-run cost rate. Iterative algorithms are provided to derive the optimal policies for both strategies. The structures of these optimal policies and their corresponding optimal cost rates are discussed and compared.

A note on bias optimality in controlled queueing systems

2000 ◽  
Vol 37 (1) ◽  
pp. 300-305 ◽  
Author(s):  
Mark E. Lewis ◽  
Martin L. Puterman
Keyword(s):  
Optimal Policy ◽  
Queueing Systems ◽  
Control Limit ◽  
Lower Control Limit ◽  
Optimal Policies ◽  
Holding Cost ◽  
Bias Optimality ◽  
Number Of Customers

The use of bias optimality to distinguish among gain optimal policies was recently studied by Haviv and Puterman [1] and extended in Lewis et al. [2]. In [1], upon arrival to an M/M/1 queue, customers offer the gatekeeper a reward R. If accepted, the gatekeeper immediately receives the reward, but is charged a holding cost, c(s), depending on the number of customers in the system. The gatekeeper, whose objective is to ‘maximize’ rewards, must decide whether to admit the customer. If the customer is accepted, the customer joins the queue and awaits service. Haviv and Puterman [1] showed there can be only two Markovian, stationary, deterministic gain optimal policies and that only the policy which uses the larger control limit is bias optimal. This showed the usefulness of bias optimality to distinguish between gain optimal policies. In the same paper, they conjectured that if the gatekeeper receives the reward upon completion of a job instead of upon entry, the bias optimal policy will be the lower control limit. This note confirms that conjecture.

On the optimal control of a deterministic epidemic

1974 ◽  
Vol 6 (04) ◽  
pp. 622-635 ◽  
Author(s):  
R. Morton ◽  
K. H. Wickwire
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Sufficient Conditions ◽  
Bounded Control ◽  
Closed Population ◽  
Control Scheme ◽  
Optimal Policies ◽  
Sufficient Conditions For Optimality ◽  
The Cost ◽  
Cost Functionals

A control scheme for the immunisation of susceptibles in the Kermack-McKendrick epidemic model for a closed population is proposed. The bounded control appears linearly in both dynamics and integral cost functionals and any optimal policies are of the “bang-bang” type. The approach uses Dynamic Programming and Pontryagin's Maximum Principle and allows one, for certain values of the cost and removal rates, to apply necessary and sufficient conditions for optimality and show that a one-switch candidate is the optimal control. In the remaining cases we are still able to show that an optimal control, if it exists, has at most one switch.

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