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.

Optimal control of service rates in networks of queues

1987 ◽  
Vol 19 (01) ◽  
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.

OPTIMAL CONTROL POLICIES FOR AN M/M/1 QUEUE WITH A REMOVABLE SERVER AND DYNAMIC SERVICE RATES

2019 ◽  
pp. 1-21
Author(s):  
Pamela Badian-Pessot ◽  
Mark E. Lewis ◽  
Douglas G. Down
Keyword(s):  
Optimal Control ◽  
Rate Control ◽  
Optimal Policy ◽  
Numerical Experiments ◽  
Service Rate ◽  
Warm Up ◽  
Cost Criterion ◽  
Optimal Service ◽  
Removable Server ◽  
Service Rates

AbstractWe consider an M/M/1 queue with a removable server that dynamically chooses its service rate from a set of finitely many rates. If the server is off, the system must warm up for a random, exponentially distributed amount of time, before it can begin processing jobs. We show under the average cost criterion, that work conserving policies are optimal. We then demonstrate the optimal policy can be characterized by a threshold for turning on the server and the optimal service rate increases monotonically with the number in system. Finally, we present some numerical experiments to provide insights into the practicality of having both a removable server and service rate control.

Rare events in series of queues

1992 ◽  
Vol 29 (1) ◽  
pp. 168-175 ◽  
Author(s):  
Pantelis Tsoucas
Keyword(s):  
Queueing Networks ◽  
Total Population ◽  
Large Deviation ◽  
Rare Event ◽  
Arrival Rate ◽  
Service Rate ◽  
Action Functional ◽  
In Series ◽  
Service Rates ◽  
Queues In Series

In an ergodic network of K M/M/1 queues in series we consider the rare event that, as N increases, the total population in the network exceeds N during a busy period. By utilizing the contraction principle of large deviation theory, an action functional is obtained for this exit problem. The ensuing minimization is carried out for K = 2 and an indication is given for arbitrary K. It is shown that, asymptotically and for unequal service rates, the ‘most likely' path for this rare event is one where the arrival rate has been interchanged with the smallest service rate. The problem has been posed in Parekh and Walrand [7] in connection with importance sampling simulation methods for queueing networks. Its solution has previously been obtained only heuristically.

scholarly journals PENERAPAN METODE ANTRIAN DALAM MENENTUKAN FASILITAS YANG OPTIMAL PADA SPBU MAWADDAH

2018 ◽  
Vol 3 (5) ◽  
Author(s):  
Diana Khairani Sofyan ◽  
Sri Meutia
Keyword(s):  
Diesel Fuel ◽  
Arrival Rate ◽  
Service Level ◽  
Service Rate ◽  
Queuing Model ◽  
Fuel Pump ◽  
Gas Stations ◽  
Scenario Design ◽  
Service Rates ◽  
Number Of Customers

Gas stations Mawaddah Is one of the gas stations located in the Village Batuphat East Lhokseumawe. The gas station has 5 oil pumps consisting of premium with two pumps, diesel consists of two pumps, and pertamax consists of one pump. Preliminary data have been made regarding the arrival rate of vehicles in each pump, which is a two-wheeled premium filling pump of 195 vehicles, four or more 166 wheels or four wheels filling pumps, four or more diesel fuel pumps of 156 and a feeding pump of 138 vehicles. High vehicle arrival rate resulted in queue. To calculate the level of service has never been done so it is not known the maximum time for service on each pump. The research method used is queuing model related to arrival rate and service level, with result of research which obtained is vehicle arrival rate at each pump that is 2 wheel of premium gasoline pump is 2.59 minutes. The premium 4 wheels charging pump is 6.98. The 4 wheelers diesel fuel pump is 5.97 minutes and the first charging pump is 6.65 minutes with the facility number 1. Vehicle service rates of premium 2 and 4 wheelers are 15.52 minutes and 14.11 minutes, 4 wheel diesel fuel pump is 14.21 minutes and the first feed pump is 13.55 minutes with scenario design on each pump is Scenario 1 with 2 pumps, Probability of medium system empty 0.87500, Number of subscribers in the system and number of customers waiting in the queue of each 1 customer, the average customer time in the system 0.06696 minutes and waiting time as long as the customer in the queue 0.00030 minutes.Keywords: Queue, facility, arrival rate, service rate.

Rare events in series of queues

1992 ◽  
Vol 29 (01) ◽  
pp. 168-175 ◽  
Author(s):  
Pantelis Tsoucas
Keyword(s):  
Queueing Networks ◽  
Total Population ◽  
Large Deviation ◽  
Rare Event ◽  
Arrival Rate ◽  
Service Rate ◽  
Action Functional ◽  
In Series ◽  
Service Rates ◽  
Queues In Series

In an ergodic network of K M/M/1 queues in series we consider the rare event that, as N increases, the total population in the network exceeds N during a busy period. By utilizing the contraction principle of large deviation theory, an action functional is obtained for this exit problem. The ensuing minimization is carried out for K = 2 and an indication is given for arbitrary K. It is shown that, asymptotically and for unequal service rates, the ‘most likely' path for this rare event is one where the arrival rate has been interchanged with the smallest service rate. The problem has been posed in Parekh and Walrand [7] in connection with importance sampling simulation methods for queueing networks. Its solution has previously been obtained only heuristically.

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.

Optimal Thresholds of an Infinite Buffer Discrete-Time Two-Server System with Triadic Policy

2011 ◽  
Vol 2 (4) ◽  
pp. 75-88
Author(s):  
Veena Goswami ◽  
G. B. Mund
Keyword(s):  
Discrete Time ◽  
Queueing System ◽  
Minimum Cost ◽  
Service Rate ◽  
Closed Form Solutions ◽  
Optimal Thresholds ◽  
Optimal Service ◽  
Number Of Customers ◽  
System Operating ◽  
Server System

This paper analyzes a discrete-time infinite-buffer Geo/Geo/2 queue, in which the number of servers can be adjusted depending on the number of customers in the system one at a time at arrival or at service completion epoch. Analytical closed-form solutions of the infinite-buffer Geo/Geo/2 queueing system operating under the triadic (0, Q N, M) policy are derived. The total expected cost function is developed to obtain the optimal operating (0, Q N, M) policy and the optimal service rate at minimum cost using direct search method. Some performance measures and sensitivity analysis have been presented.

Convexity results for single-server queues and for multiserver queues with constant service times

1990 ◽  
Vol 27 (02) ◽  
pp. 465-468 ◽  
Author(s):  
Arie Harel
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Sojourn Time ◽  
Interarrival Time ◽  
Service Rate ◽  
Single Server ◽  
Multiserver Queues ◽  
Service Rates ◽  
Constant Service Times ◽  
Number Of Customers

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates. These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

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.

Throughput properties of a queueing network with distributed dynamic routing and flow control

1996 ◽  
Vol 28 (01) ◽  
pp. 285-307 ◽  
Author(s):  
Leandros Tassiulas ◽  
Anthony Ephremides
Keyword(s):  
Flow Control ◽  
Rate Control ◽  
Queueing Network ◽  
Local State ◽  
Service Rate ◽  
Maximum Throughput ◽  
State Information ◽  
Arbitrary Topology ◽  
State Dependent ◽  
Service Rates

A queueing network with arbitrary topology, state dependent routing and flow control is considered. Customers may enter the network at any queue and they are routed through it until they reach certain queues from which they may leave the system. The routing is based on local state information. The service rate of a server is controlled based on local state information as well. A distributed policy for routing and service rate control is identified that achieves maximum throughput. The policy can be implemented without knowledge of the arrival and service rates. The importance of flow control is demonstrated by showing that, in certain networks, if the servers cannot be forced to idle, then no maximum throughput policy exists when the arrival rates are not known. Also a model for exchange of state information among neighboring nodes is presented and the network is studied when the routing is based on delayed state information. A distributed policy is shown to achieve maximum throughput in the case of delayed state information. Finally, some implications for deterministic flow networks are discussed.

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