scholarly journals SCHEDULING IN A SINGLE-SERVER QUEUE WITH STATE-DEPENDENT SERVICE RATES

2019 ◽  
Vol 34 (4) ◽  
pp. 507-521
Author(s):  
Urtzi Ayesta ◽  
Balakrishna Prabhu ◽  
Rhonda Righter
Keyword(s):  
Optimal Policy ◽  
Arrival Rate ◽  
Complete Characterization ◽  
Single Server ◽  
Holding Costs ◽  
Release Times ◽  
State Dependent ◽  
Service Rates ◽  
Transient System ◽  
The Cost

We consider single-server scheduling to minimize holding costs where the capacity, or rate of service, depends on the number of jobs in the system, and job sizes become known upon arrival. In general, this is a hard problem, and counter-intuitive behavior can occur. For example, even with linear holding costs the optimal policy may be something other than SRPT or LRPT, it may idle, and it may depend on the arrival rate. We first establish an equivalence between our problem of deciding which jobs to serve when completed jobs immediately leave, and a problem in which we have the option to hold on to completed jobs and can choose when to release them, and in which we always serve jobs according to SRPT. We thus reduce the problem to determining the release times of completed jobs. For the clearing, or transient system, where all jobs are present at time 0, we give a complete characterization of the optimal policy and show that it is fully determined by the cost-to-capacity ratio. With arrivals, the problem is much more complicated, and we can obtain only partial results.

Optimal control of service rates in networks of queues

1987 ◽  
Vol 19 (1) ◽  
pp. 202-218 ◽  
Author(s):  
Richard R. Weber ◽  
Shaler Stidham
Keyword(s):  
Rate Control ◽  
Queueing Networks ◽  
Arrival Rate ◽  
Service Rate ◽  
Discounted Cost ◽  
Holding Costs ◽  
Optimal Service ◽  
Monotonicity Result ◽  
Service Rates ◽  
Number Of Customers

We prove a monotonicity result for the problem of optimal service rate control in certain queueing networks. Consider, as an illustrative example, a number of ·/M/1 queues which are arranged in a cycle with some number of customers moving around the cycle. A holding cost hi(xi) is charged for each unit of time that queue i contains xi customers, with hi being convex. As a function of the queue lengths the service rate at each queue i is to be chosen in the interval , where cost ci(μ) is charged for each unit of time that the service rate μis in effect at queue i. It is shown that the policy which minimizes the expected total discounted cost has a monotone structure: namely, that by moving one customer from queue i to the following queue, the optimal service rate in queue i is not increased and the optimal service rates elsewhere are not decreased. We prove a similar result for problems of optimal arrival rate and service rate control in general queueing networks. The results are extended to an average-cost measure, and an example is included to show that in general the assumption of convex holding costs may not be relaxed. A further example shows that the optimal policy may not be monotone unless the choice of possible service rates at each queue includes 0.

scholarly journals Service Rate Optimization of Finite Population Queueing Model with State Dependent Arrival and Service Rates

2018 ◽  
Vol 13 (1) ◽  
pp. 60-68
Author(s):  
Sushil Ghimire ◽  
Gyan Bahadur Thapa ◽  
Ram Prasad Ghimire
Keyword(s):  
Finite Population ◽  
Arrival Rate ◽  
Service Rate ◽  
Queueing Model ◽  
State Dependent ◽  
Service Rates ◽  
Rate Optimization

 Providing service immediately after the arrival is rarely been used in practice. But there are some situations for which servers are more than the arrivals and no one has to wait to get served. In this model, arrival rate is

scholarly journals Optimal stochastic scheduling of forest networks with switching penalties

1994 ◽  
Vol 26 (02) ◽  
pp. 474-497 ◽  
Author(s):  
Mark P. Van Oyen ◽  
Demosthenis Teneketzis
Keyword(s):  
Structural Properties ◽  
Optimal Policy ◽  
Switching Costs ◽  
Stochastic Scheduling ◽  
Single Server ◽  
Reward Rate ◽  
Holding Costs ◽  
Lump Sum ◽  
Optimal Policies ◽  
Set Up

We present structural properties of optimal policies for the problem of scheduling a single server in a forest network of N queues (without arrivals) subject to switching penalties. In addition to linear holding costs, we impose either lump sum switching costs or batch set-up delays which are incurred at each instant the server processes a job in a queue different from the previous one. We use reward rate notions to unearth conditions on the holding costs and service distributions for which an exhaustive policy is optimal. For the case of two nodes connected probabilistically in tandem, we explicitly define an optimal policy under similar conditions.

scholarly journals Tax problems in the undiscounted case

2005 ◽  
Vol 42 (3) ◽  
pp. 754-765 ◽  
Author(s):  
P. Whittle
Keyword(s):  
Optimal Policy ◽  
The State ◽  
Gittins Index ◽  
Index Policy ◽  
State Dependent ◽  
Dependent Part ◽  
Occupation Numbers ◽  
The Matrix ◽  
The Cost ◽  
Programming Analysis

The aim of this paper is to evaluate the performance of the optimal policy (the Gittins index policy) for open tax problems of the type considered by Klimov in the undiscounted limit. In this limit, the state-dependent part of the cost is linear in the state occupation numbers for the multi-armed bandit, but is quadratic for the tax problem. The discussion of the passage to the limit for the tax problem is believed to be largely new; the principal novelty is our evaluation of the matrix of the quadratic form. These results are confirmed by a dynamic programming analysis, which also suggests how the optimal policy should be modified when resources can be freely deployed only within workstations, rather than system-wide.

scholarly journals Optimal stochastic scheduling of forest networks with switching penalties

1994 ◽  
Vol 26 (2) ◽  
pp. 474-497 ◽  
Author(s):  
Mark P. Van Oyen ◽  
Demosthenis Teneketzis
Keyword(s):  
Structural Properties ◽  
Optimal Policy ◽  
Switching Costs ◽  
Stochastic Scheduling ◽  
Single Server ◽  
Reward Rate ◽  
Holding Costs ◽  
Lump Sum ◽  
Optimal Policies ◽  
Set Up

We present structural properties of optimal policies for the problem of scheduling a single server in a forest network of N queues (without arrivals) subject to switching penalties. In addition to linear holding costs, we impose either lump sum switching costs or batch set-up delays which are incurred at each instant the server processes a job in a queue different from the previous one. We use reward rate notions to unearth conditions on the holding costs and service distributions for which an exhaustive policy is optimal. For the case of two nodes connected probabilistically in tandem, we explicitly define an optimal policy under similar conditions.

DYNAMIC CONTROL OF A SINGLE-SERVER SYSTEM WHEN JOBS CHANGE STATUS

2017 ◽  
Vol 32 (3) ◽  
pp. 353-395
Author(s):  
Gabriel Zayas-Cabán ◽  
Hyun-Soo Ahn
Keyword(s):  
Queueing Systems ◽  
Dynamic Control ◽  
Sufficient Conditions ◽  
Single Server ◽  
Priority Rules ◽  
Holding Costs ◽  
Class 1 ◽  
Markov Decision ◽  
Service Rates ◽  
Customer Model

From health care to maintenance shops, many systems must contend with allocating resources to customers or jobs whose initial service requirements or costs change when they wait too long. We present a new queueing model for this scenario and use a Markov decision process formulation to analyze assignment policies that minimize holding costs. We show that the classic cμ rule is generally not optimal when service or cost requirements can change. Even for a two-class customer model where a class 1 task becomes a class 2 task upon waiting, we show that additional orderings of the service rates are needed to ensure the optimality of simple priority rules. We then show that seemingly-intuitive switching curve structures are also not optimal in general. We study these scenarios and provide conditions under which they do hold. Lastly, we show that results from the two-class model do not extend to when there are n≥3 customer classes. More broadly, we find that simple priority rules are not optimal. We provide sufficient conditions under which a simple priority rule holds. In short, allowing service and/or cost requirements to change fundamentally changes the structure of the optimal policy for resource allocation in queueing systems.

scholarly journals SCHEDULING IMPATIENT JOBS IN A CLEARING SYSTEM WITH INSIGHTS ON PATIENT TRIAGE IN MASS CASUALTY INCIDENTS

2008 ◽  
Vol 22 (3) ◽  
pp. 301-332 ◽  
Author(s):  
Nilay Tanik Argon ◽  
Serhan Ziya ◽  
Rhonda Righter
Keyword(s):  
Service Time ◽  
Optimal Policy ◽  
Numerical Study ◽  
Complex Structure ◽  
Mass Casualty ◽  
Single Server ◽  
Worst Case ◽  
State Dependent ◽  
Clearing System ◽  
Mass Casualty Event

Motivated by the patient triage problem in emergency response, we consider a single-server clearing system in which jobs might abandon the system if they are not taken into service within their “lifetime.” In this system, jobs are characterized by their lifetime and service time distributions. Our objective is to dynamically determine the optimal or near-optimal order of service for jobs so as to minimize the total number of abandonments. We first show that if the jobs can be ordered in such a way that the job with the shortest lifetime (in the sense of hazard rate ordering) also has the shortest service time (in the sense of likelihood ratio ordering), then the optimal policy gives the highest priority to this “time-critical” job independently of the system state. For the case in which the jobs with shorter lifetimes have longer service times, we observed that the optimal policy generally has a complex structure that might depend on the type and number of jobs available. For this case, we provide partial characterizations of the optimal policy and obtain sufficient conditions under which a state-independent policy is optimal. Furthermore, we develop two state-dependent heuristic policies, and by means of a numerical study, we show that these heuristics perform well, especially when jobs abandon the system at a relatively faster rate when compared to service rates. Based on our analytical and numerical results, we develop several insights on patient triage in the immediate aftermath of a mass casualty event. For example, we conclude that in a worst-case scenario, where medical resources are overwhelmed with a large number of casualties who need immediate attention, it is crucial to implement state-dependent policies such as the heuristic policies proposed in this article.

scholarly journals OPTIMAL SWITCHING ON AND OFF THE ENTIRE SERVICE CAPACITY OF A PARALLEL QUEUE

2015 ◽  
Vol 29 (4) ◽  
pp. 483-506 ◽  
Author(s):  
Eugene A. Feinberg ◽  
Xiaoxuan Zhang
Keyword(s):  
Linear Programming ◽  
Finite Number ◽  
Optimal Policy ◽  
Average Cost ◽  
Optimization Problem ◽  
System Size ◽  
Single Server ◽  
Service Capacity ◽  
Optimal Switching ◽  
Holding Costs

This paper studies optimal switching on and off of the entire service capacity of an M/M/∞ queue with holding, running and switching costs. The running costs depend only on whether the system is on or off, and the holding costs are linear. The goal is to minimize average costs per unit time. The main result is that an average-cost optimal policy either always runs the system or is an (M, N)-policy defined by two thresholds M and N, such that the system is switched on upon an arrival epoch when the system size accumulates to N and is switched off upon a departure epoch when the system size decreases to M. It is shown that this optimization problem can be reduced to a problem with a finite number of states and actions, and an average-cost optimal policy can be computed via linear programming. An example, in which the optimal (M, N)-policy outperforms the best (0, N)-policy, is provided. Thus, unlike the case of single-server queues studied in the literature, (0, N)-policies may not be average-cost optimal.

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