On the Assignment of Customers to Parallel Queues

1992 ◽  
Vol 6 (4) ◽  
pp. 495-511 ◽  
Author(s):  
Arie Hordijk ◽  
Ger Koole
Keyword(s):  
Dynamic Programming ◽  
Cost Function ◽  
Optimal Policy ◽  
The State ◽  
Arrival Process ◽  
Single Server ◽  
Finite Buffers ◽  
Parallel Queues ◽  
Routing Policy ◽  
The Cost

This paper considers routing to parallel queues in which each queue has its own single server and service times are exponential with nonidentical parameters. We give conditions on the cost function such that the optimal policy assigns customers to a faster queue when that server has a shorter queue. The queues may have finite buffers, and the arrival process can be controlled and can depend on the state and routing policy. Hence our results on the structure of the optimal policy are also true when the assigning control is in the “last” node of a network of service centers. Using dynamic programming we show that our optimality results are true in distribution.

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.

ON PARALLEL QUEUING WITH RANDOM SERVER CONNECTIVITY AND ROUTING CONSTRAINTS

2002 ◽  
Vol 16 (2) ◽  
pp. 185-203 ◽  
Author(s):  
Nicholas Bambos ◽  
George Michailidis
Keyword(s):  
Classical Model ◽  
Single Server ◽  
Job Flows ◽  
Queuing Model ◽  
Finite Buffers ◽  
Parallel Queues ◽  
The Third ◽  
Buffer Overflows ◽  
Random Connectivity ◽  
Application Scope

We study systems of parallel queues with finite buffers, a single server with random connectivity to each queue, and arriving job flows with random or class-dependent accessibility to the queues. Only currently connected queues may receive (preemptive) service at any given time, whereas an arriving job can only join one of its accessible queues. Using the coupling method, we study three key models, progressively building from simpler to more complicated structures.In the first model, there are only random server connectivities. It is shown that allocating the server to the Connected queue with the Fewest Empty Spaces (C-FES) stochastically minimizes the number of lost jobs due to buffer overflows, under conditions of independence and symmetry.In the second model, we additionally consider random accessibility of queues by arriving jobs. It is shown that allocating the server to the C-FES and routing each arriving job to the currently Accessible queue with the Most Empty Spaces (C-FES/A-MES) minimizes the loss flow stochastically, under similar assumptions.In the third model (addressing a target application), we consider multiple classes of arriving job flows, each allowed access to a deterministic subset of the queues. Under analogous assumptions, it is again shown that the C-FES/A-MES policy minimizes the loss flow stochastically.The random connectivity/accessibility aspect enhances significantly the structure and application scope of the classical parallel queuing model. On the other hand, it introduces essential additional dynamics and considerable complications. It is interesting that a simple policy like FES/MES, known to be optimal for the classical model, extends to the C-FES/A-MES in our case.

Energy Cost Optimization of HVAC Loads Under Time-Varying Electricity Price Signals

2016 ◽  
Author(s):  
Saeid Bashash
Keyword(s):  
Dynamic Programming ◽  
Cost Function ◽  
Energy Cost ◽  
Optimal Operation ◽  
Electricity Price ◽  
Programming Approach ◽  
Time Varying ◽  
Temperature Deviation ◽  
Price Signals ◽  
The Cost

This paper presents a dynamic programming approach to optimize energy cost of multiple interacting household appliances such as air conditioning systems and refrigerators with temperature flexibility, under time varying electricity price signals. We adopt a first order differential equation model with a binary (ON-OFF) switching control function for each load. An energy cost minimization problem is then formulated with a pair of constraints on the temperature lower and upper bounds, as well as an equality condition on the initial and final temperature states. We use dynamic programming to compute cost-optimal control inputs and temperature trajectories for a given electricity price profile and ambient temperature condition. To account for temperature deviation from its desired setpoint, a quadratic temperature deviation penalty is added to the cost function. Moreover, to minimize the control input chattering for equipment protection, the cost function is expanded to also minimize the number of on-off switching events. Results for the different weighting combinations of the optimization objectives provide useful insights on the optimal operation of individual and multiple interacting HVAC loads. In particular, we observe that the loads are desynchronized under the cost-optimal operation, in the presence of local (renewable) power generation. The presented optimization algorithm and observed results can lead to the development of novel model predictive and rule-based feedback control policies for optimal energy management in households.

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.

Scheduling jobs on non-identical IFR processors to minimize general cost functions

1991 ◽  
Vol 23 (04) ◽  
pp. 909-924 ◽  
Author(s):  
Rhonda Righter ◽  
Susan H. Xu
Keyword(s):  
Cost Function ◽  
Optimal Policy ◽  
Expected Number ◽  
Expected Cost ◽  
Processing Times ◽  
Special Cases ◽  
The Cost ◽  
General Cost Functions ◽  
Tardy Jobs ◽  
Completion Times

We consider the problem of scheduling n jobs non-preemptively on m parallel, non-identical processors to minimize a weighted expected cost function of job completion times, where the weights are associated with the jobs. The cost function is assumed to be increasing and concave but otherwise arbitrary. Processing times are IFR with different distributions for different processors. Jobs may be processed on any processor and there are no precedences. We show that the optimal policy orders the jobs in decreasing order of their weights and then uses the individually optimal policy for each job. In other words, processors are offered to jobs in order, and each job considers its own expected cost function for its completion time to decide whether to accept or reject a processor. Therefore, the optimal policy does not depend on the weights of the jobs except through their order. Special cases of our objective function are weighted expected flowtime, weighted discounted expected flowtime, and weighted expected number of tardy jobs.

scholarly journals Equilibrium Customer Strategies in the Single-Server Constant Retrial Queue with Breakdowns and Repairs

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Zhengwu Zhang ◽  
Jinting Wang ◽  
Feng Zhang
Keyword(s):  
Retrial Queue ◽  
Profit Maximization ◽  
Optimal Strategies ◽  
The State ◽  
Equilibrium Analysis ◽  
Arrival Process ◽  
Single Server ◽  
Retrial Queueing System ◽  
Exact Number ◽  
Number Of Customers

We consider a single-server constant retrial queueing system with a Poisson arrival process and exponential service and retrial times, in which the server may break down when it is working. The lifetime of the server is assumed to be exponentially distributed and once the server breaks down, it will be sent for repair immediately and the repair time is also exponentially distributed. There is no waiting space in front of the server and arriving customers decide whether to enter the retrial orbit or to balk depending on the available information they get upon arrival. In the paper, Nash equilibrium analysis for customers’ joining strategies as well as the related social and profit maximization problems is investigated. We consider separately the partially observable case where an arriving customer knows the state of the server but does not observe the exact number of customers waiting for service and the fully observable case where customer gets informed not only about the state of the server but also about the exact number of customers in the orbit. Some numerical examples are presented to illustrate the effect of the information levels and several parameters on the customers’ equilibrium and optimal strategies.

Scheduling jobs on non-identical IFR processors to minimize general cost functions

1991 ◽  
Vol 23 (4) ◽  
pp. 909-924 ◽  
Author(s):  
Rhonda Righter ◽  
Susan H. Xu
Keyword(s):  
Cost Function ◽  
Optimal Policy ◽  
Expected Number ◽  
Expected Cost ◽  
Processing Times ◽  
Special Cases ◽  
The Cost ◽  
General Cost Functions ◽  
Tardy Jobs ◽  
Completion Times

We consider the problem of scheduling n jobs non-preemptively on m parallel, non-identical processors to minimize a weighted expected cost function of job completion times, where the weights are associated with the jobs. The cost function is assumed to be increasing and concave but otherwise arbitrary. Processing times are IFR with different distributions for different processors. Jobs may be processed on any processor and there are no precedences. We show that the optimal policy orders the jobs in decreasing order of their weights and then uses the individually optimal policy for each job. In other words, processors are offered to jobs in order, and each job considers its own expected cost function for its completion time to decide whether to accept or reject a processor. Therefore, the optimal policy does not depend on the weights of the jobs except through their order. Special cases of our objective function are weighted expected flowtime, weighted discounted expected flowtime, and weighted expected number of tardy jobs.

Applying Dynamic Programming Algorithms to the Energy Management of Hybrid Electric Aircraft

2018 ◽  
Author(s):  
Teresa Donateo ◽  
Antonio Ficarella ◽  
Luigi Spedicato
Keyword(s):  
Dynamic Programming ◽  
Fuel Consumption ◽  
Energy Management ◽  
The State ◽  
Light Transport ◽  
Local Optimum ◽  
Dijkstra Algorithm ◽  
Electric Aircraft ◽  
Hybrid Electric ◽  
The Cost

To explore the application of dynamic programming (DP) to the energy management strategies of hybrid electric aircraft, a hybrid powertrain for a lightweight rotorcraft is introduced and its dynamic control model is designed. The model is conceived for the Agusta-Westland A109 helicopter, a twin-engine rotorcraft used in various roles, such as light transport, search-and-rescue and military roles. The turboshaft single spool engines are modeled with the use of performance maps that allow part load specific fuel consumption to be calculated as a function of actual power request and flight conditions. The state-of-the-art lithium polymer batteries are used for the hybridization and their behavior is evaluated by the Sheperd-Peukert model. The control problem is modeled through a graph structure where a node is obtained from the intersection between a time value, representing the starting of a phase of flight, and a splitting factor, representing the percentage of propulsive power required to the battery in such a phase. The edge connecting two nodes concerns with the state transition and the weight of the edge refers to the transition cost. The goal is to find an optimal splitting sequence to minimize the total cost over the whole mission, that is given with regard to speed and altitude. The Dijkstra algorithm, which allows the shortest energy path to be found between nodes in a graph, is used to look for the optimum. A local optimum is achieved when the cost is defined as the fuel consumption whereas the global optimum can be attained when the model is enhanced to include the effect of the battery usage into the cost. The results are compared with the original non-hybrid case and the engine efficiency was suitable evaluated. The applicability to other mission data is suitably evaluated so as to deduce the concept of similarity of mission.

scholarly journals Algorithm for Solving Problem of Designing Regional Logistics Infrastructure

2021 ◽  
Vol 20 (4) ◽  
pp. 352-356
Author(s):  
R. B. Ivut ◽  
P. V. Popov ◽  
P. I. Lapkovskaya ◽  
N. E. Sheveleva
Keyword(s):  
Dynamic Programming ◽  
Cost Function ◽  
Methodological Approach ◽  
Optimal Location ◽  
Transport Infrastructure ◽  
Regional Logistics ◽  
Initial Stage ◽  
Nested Sets ◽  
Three Stages ◽  
The Cost

The paper considers an algorithm for solving the problem of finding the optimal location of key objects of transport and warehouse infrastructures within the framework of a methodological approach to designing logistics infrastructure in the territory of the region of the countries. The methodological approach includes three stages. At the first stage, areas  are determined where it is advisable to locate key objects of the regional logistics infrastructure. Further, using the models developed by the authors, the linking of warehouse infrastructure objects on the ground has been carried out and, taking into account the designed warehouse network, the optimal dislocation of transport infrastructure objects has been determined.  To find the optimal locations for the objects for regional logistics infrastructure facilities, the authors propose an algorithm that is applicable both for building warehouse and transport infrastructures due to the similarity of the models. The algorithm is based on the method of constructing a sequence of plans. At the initial stage, the final expansion is constructed for the set of plans under consideration. For a given set, a minorant has been determined for the cost function associated with the placement and maintenance of infrastructure facilities, the movement of goods, and the haul of an empty vehicle. After that, an iterative algorithm has been formed that determines the sequence of optima of the minorant on a sequence of nested sets. At the first step, an element of the set of plans has been found that minimizes the minorant, at the next step, the found element is excluded from the set under consideration, and a new optimum is sought on the remaining set for which the minorant takes the minimum value. To eliminate multiple plans, it is advisable to use dynamic programming procedures. The limits of applicability  of the method for constructing a sequence of plans are determined by the ability to construct an extension of the set of plans for placing objects, select a minorant on it, and build an algorithm for ordering optima.

OPTIMAL CONTROL OF PARALLEL QUEUES WITH BATCH SERVICE

2002 ◽  
Vol 16 (3) ◽  
pp. 289-307 ◽  
Author(s):  
Cathy H. Xia ◽  
George Michailidis ◽  
Nicholas Bambos ◽  
Peter W. Glynn
Keyword(s):  
Fluid Model ◽  
Heavy Traffic ◽  
Batch Size ◽  
Single Server ◽  
Dynamic Allocation ◽  
Finite Buffers ◽  
Threshold Policy ◽  
Parallel Queues ◽  
Buffer Overflows ◽  
Arrival Processes

We consider the problem of dynamic allocation of a single server with batch processing capability to a set of parallel queues. Jobs from different classes cannot be processed together in the same batch. The arrival processes are mutually independent Poisson flows with equal rates. Batches have independent and identically distributed exponentially distributed service times, independent of the batch size and the arrival processes. It is shown that for the case of infinite buffers, allocating the server to the longest queue, stochastically maximizes the aggregate throughput of the system. For the case of equal-size finite buffers the same policy stochastically minimizes the loss of jobs due to buffer overflows. Finally, for the case of unequal-size buffers, a threshold-type policy is identified through an extensive simulation study and shown to consistently outperform other conventional policies. The good performance of the proposed threshold policy is confirmed in the heavy-traffic regime using a fluid model.

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