Two competing queues with linear costs and geometric service requirements: the μc-rule is often optimal

1985 ◽  
Vol 17 (1) ◽  
pp. 186-209 ◽  
Author(s):  
J. S. Baras ◽  
A. J. Dorsey ◽  
A. M. Makowski
Keyword(s):  
State Space Model ◽  
Sample Path ◽  
Single Server ◽  
Time Model ◽  
Programming Methodology ◽  
Sample Path Properties ◽  
Long Run ◽  
Markov Decision ◽  
The Cost ◽  
Discounted Criterion

A discrete-time model is presented for a system of two queues competing for the service attention of a single server with infinite buffer capacity. The service requirements are geometrically distributed and independent from customer to customer as well as from the arrivals. The allocation of service attention is governed by feedback policies which are based on past decisions and buffer content histories. The cost of operation per unit time is a linear function of the queue sizes. Under the model assumptions, a fixed prioritization scheme, known as the μc-rule, is shown to be optimal for the expected long-run average criterion and for the expected discounted criterion, over both finite and infinite horizons. Two different approaches are proposed for solving these problems. One is based on the dynamic programming methodology for Markov decision processes, and assumes the arrivals to be i.i.d. The other is valid under no additional assumption on the arrival stream and uses direct comparison arguments. In both cases, the sample path properties of the adopted state-space model are exploited.

Two competing queues with linear costs and geometric service requirements: the μc-rule is often optimal

1985 ◽  
Vol 17 (01) ◽  
pp. 186-209
Author(s):  
J. S. Baras ◽  
A. J. Dorsey ◽  
A. M. Makowski
Keyword(s):  
State Space Model ◽  
Sample Path ◽  
Single Server ◽  
Time Model ◽  
Programming Methodology ◽  
Sample Path Properties ◽  
Long Run ◽  
Markov Decision ◽  
The Cost ◽  
Discounted Criterion

A discrete-time model is presented for a system of two queues competing for the service attention of a single server with infinite buffer capacity. The service requirements are geometrically distributed and independent from customer to customer as well as from the arrivals. The allocation of service attention is governed by feedback policies which are based on past decisions and buffer content histories. The cost of operation per unit time is a linear function of the queue sizes. Under the model assumptions, a fixed prioritization scheme, known as the μc-rule, is shown to be optimal for the expected long-run average criterion and for the expected discounted criterion, over both finite and infinite horizons. Two different approaches are proposed for solving these problems. One is based on the dynamic programming methodology for Markov decision processes, and assumes the arrivals to be i.i.d. The other is valid under no additional assumption on the arrival stream and uses direct comparison arguments. In both cases, the sample path properties of the adopted state-space model are exploited.

Two competing queues with linear costs: the μc-rule is often optimal

1984 ◽  
Vol 16 (1) ◽  
pp. 8-8
Author(s):  
J. S. Baras ◽  
A. J. Dorsey ◽  
A. M. Makowski
Keyword(s):  
State Space ◽  
Random Sequence ◽  
State Space Model ◽  
Sample Path ◽  
Single Server ◽  
Space Model ◽  
Long Run ◽  
Markov Decision ◽  
Cost Parameters ◽  
Discounted Criterion

A state-space model is presented for a queueing system where two classes of customer compete in discrete-time for the service attention of a single server with infinite buffer capacity. The arrivals are modelled by an independent identically distributed random sequence of a general type while the service completions are generated by independent Bernoulli streams; the allocation of service attention is governed by feedback policies which are based on past decisions and buffer content histories. The cost of operation per unit time is a linear function of the queue sizes. Under the model assumptions, a fixed prioritization scheme, known as the μc -rule, is shown to be optimal when the expected long-run average criterion and the expected discounted criterion, over both finite and infinite horizons, are used. This static prioritization of the two classes of customers is done solely on the basis of service and cost parameters. The analysis is based on the dynamic programming methodology for Markov decision processes and takes advantage of the sample-path properties of the adopted state-space model.

scholarly journals Large Deviations for the Single-Server Queue and the Reneging Paradox

2021 ◽  
Author(s):  
Rami Atar ◽  
Amarjit Budhiraja ◽  
Paul Dupuis ◽  
Ruoyu Wu
Keyword(s):  
Large Deviations ◽  
Decay Rate ◽  
Sample Path ◽  
Arrival Rate ◽  
Rate Function ◽  
Service Rate ◽  
Single Server ◽  
Large Deviations Principle ◽  
Long Run ◽  
The Individual

For the M/M/1+M model at the law-of-large-numbers scale, the long-run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: the decay rate of the probability that the long-run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.

scholarly journals OPTIMAL ADMISSION AND ROUTING WITH CONGESTION-SENSITIVE CUSTOMER CLASSES

2021 ◽  
pp. 1-25
Author(s):  
Ayse Aslan
Keyword(s):  
Sample Path ◽  
Service Station ◽  
Holding Costs ◽  
Long Run ◽  
Optimal Policies ◽  
Markov Decision ◽  
Alternative Service ◽  
Routing Control ◽  
The Difference ◽  
Self Service

This paper considers optimal admission and routing control in multi-class service systems in which customers can either receive quality regular service which is subject to congestion or can receive congestion-free but less desirable service at an alternative service station, which we call the self-service station. We formulate the problem within the Markov decision process framework and focus on characterizing the structure of dynamic optimal policies which maximize the expected long-run rewards. For this, value function and sample path arguments are used. The congestion sensitivity of customers is modeled with class-independent holding costs at the regular service station. The results show how the admission rewards of customer classes affect their priorities at the regular and self-service stations. We explore that the priority for regular service may not only depend on regular service admission rewards of classes but also on the difference between regular and self-service admission rewards. We show that optimal policies have monotonicity properties, regarding the optimal decisions of individual customer classes such that they divide the state space into three connected regions per class.

scholarly journals DYNAMIC ROUTING OF CUSTOMERS WITH GENERAL DELAY COSTS IN A MULTISERVER QUEUING SYSTEM

2009 ◽  
Vol 23 (2) ◽  
pp. 175-203 ◽  
Author(s):  
Nilay Tanik Argon ◽  
Li Ding ◽  
Kevin D. Glazebrook ◽  
Serhan Ziya
Keyword(s):  
Lagrangian Relaxation ◽  
Numerical Study ◽  
Heavy Traffic ◽  
Significant Advance ◽  
Greedy Heuristic ◽  
Single Server ◽  
Long Run ◽  
Central Controller ◽  
Routing Policies ◽  
The Cost

We consider a network of parallel service stations each modeled as a single-server queue. Each station serves its own dedicated customers as well as generic customers who are routed from a central controller. We suppose that the cost incurred by a customer is an increasing function of her time spent in the system. In a significant advance on most previous work, we do not require waiting costs to be convex, still less linear. With the objective of minimizing the long-run average waiting cost, we develop two heuristic routing policies, one of which is based on dynamic programming policy improvement and the other on Lagrangian relaxation. In developing the latter policy, we show that each station is “indexable” under mild conditions for customers’ waiting costs and also prove some structural results on the admission control problem that naturally arises as a result of the Lagrangian relaxation. We then test the performance of our heuristics in an extensive numerical study and show that the Lagrangian heuristic demonstrates a strong level of performance in a range of traffic conditions. In particular, it clearly outperforms both a greedy heuristic, which is a standard proposal in complex routing problems, and a recent proposal from the heavy traffic literature.

scholarly journals Application of the theory of decision making under risk and uncertainty at modelling of costs

2010 ◽  
Vol 56 (No. 5) ◽  
pp. 201-208 ◽  
Author(s):  
M. Beranová ◽  
D. Martinovičová
Keyword(s):  
Decision Making ◽  
Financial Accounting ◽  
Cost Functions ◽  
Risk And Uncertainty ◽  
Managerial Decision ◽  
Managerial Decision Making ◽  
Bayesian Theorem ◽  
Long Run ◽  
Short Run ◽  
The Cost

The costs functions are mentioned mostly in the relation to the Break-even Analysis where they are presented in the linear form. But there exist several different types and forms of cost functions. Fist of all, it is necessary to distinguish between the short-run and long-run cost function that are both very important tools of the managerial decision making even if each one is used on a different level of management. Also several methods of estimation of the cost function's parameters are elaborated in the literature. But all these methods are based on the past data taken from the financial accounting while the financial accounting is not able to separate the fixed and variable costs and it is also strongly adjusted to taxation in the many companies. As a tool of the managerial decision making support, the cost functions should provide a vision to the future where many factors of risk and uncertainty influence economic results. Consequently, these random factors should be considered in the construction of cost functions, especially in the long-run. In order to quantify the influences of these risks and uncertainties, the authors submit the application of the Bayesian Theorem.

scholarly journals Competing Conventions with Costly Information Acquisition

Games ◽  
2021 ◽  
Vol 12 (3) ◽  
pp. 53
Author(s):  
Roberto Rozzi
Keyword(s):  
Information Acquisition ◽  
Evolutionary Model ◽  
Equilibrium Selection ◽  
The Other ◽  
Group Interactions ◽  
Social Coordination ◽  
Long Run ◽  
Stochastic Stability Analysis ◽  
The Stability ◽  
The Cost

We consider an evolutionary model of social coordination in a 2 × 2 game where two groups of players prefer to coordinate on different actions. Players can pay a cost to learn their opponent’s group: if they pay it, they can condition their actions concerning the groups. We assess the stability of outcomes in the long run using stochastic stability analysis. We find that three elements matter for the equilibrium selection: the group size, the strength of preferences, and the information’s cost. If the cost is too high, players never learn the group of their opponents in the long run. If one group is stronger in preferences for its favorite action than the other, or its size is sufficiently large compared to the other group, every player plays that group’s favorite action. If both groups are strong enough in preferences, or if none of the groups’ sizes is large enough, players play their favorite actions and miscoordinate in inter-group interactions. Lower levels of the cost favor coordination. Indeed, when the cost is low, in inside-group interactions, players always coordinate on their favorite action, while in inter-group interactions, they coordinate on the favorite action of the group that is stronger in preferences or large enough.

Optimal stationary strategies in leavable Markov decision processes

1990 ◽  
Vol 27 (01) ◽  
pp. 134-145
Author(s):  
Matthias Fassbender
Keyword(s):  
Transition Probabilities ◽  
Practical Applications ◽  
Stationary Strategies ◽  
Countable State ◽  
Markov Decision ◽  
Bias Optimality ◽  
The Mean ◽  
Optimal Stationary Strategies ◽  
The Cost ◽  
Reward Criterion

This paper establishes the existence of an optimal stationary strategy in a leavable Markov decision process with countable state space and undiscounted total reward criterion. Besides assumptions of boundedness and continuity, an assumption is imposed on the model which demands the continuity of the mean recurrence times on a subset of the stationary strategies, the so-called ‘good strategies'. For practical applications it is important that this assumption is implied by an assumption about the cost structure and the transition probabilities. In the last part we point out that our results in general cannot be deduced from related works on bias-optimality by Dekker and Hordijk, Wijngaard or Mann.

Sample path behaviour of Χ2 surfaces at extrema

1988 ◽  
Vol 20 (4) ◽  
pp. 719-738 ◽  
Author(s):  
Michael Aronowich ◽  
Robert J. Adler
Keyword(s):  
Rough Surfaces ◽  
Sample Path ◽  
Gaussian Field ◽  
Random Surfaces ◽  
Sample Path Properties ◽  
Low Levels ◽  
Path Properties

We study the sample path properties of χ2 random surfaces, in particular in the neighbourhood of their extrema. We show that, as is the case for their Gaussian counterparts, χ2 surfaces at high levels follow the form of certain deterministic paraboloids, but that, unlike their Gaussian counterparts, at low levels their form is much more random. This has a number of interesting implications in the modelling of rough surfaces and the study of the ‘robustness' of Gaussian field models. The general approach of the paper is the study of extrema via the ‘Slepian model process', which, for χ2 fields, is tractable only at asymptotically high or low levels.

Sign in / Sign up

Export Citation Format

Share Document