scholarly journals On the Nash equilibria for the FCFS queueing system with load-increasing service rate

2005 ◽  
Vol 37 (2) ◽  
pp. 461-481 ◽  
Author(s):  
A. C. Brooms
Keyword(s):  
Queue Length ◽  
Nash Equilibria ◽  
Decision Rules ◽  
Learning Rule ◽  
Service Rate ◽  
Simulation Experiments ◽  
Player Game ◽  
Expected Sojourn Time ◽  
Increasing Function ◽  
Fcfs Discipline

We consider a service system (QS) that operates according to the first-come-first-served (FCFS) discipline, and in which the service rate is an increasing function of the queue length. Customers arrive sequentially at the system, and decide whether or not to join using decision rules based upon the queue length on arrival. Each customer is interested in selecting a rule that meets a certain optimality criterion with regard to their expected sojourn time in the system; as a consequence, the decision rules of other customers must be taken into account. Within a particular class of decision rules for an associated infinite-player game, the structure of the Nash equilibrium routeing policies is characterized. We prove that, within this class, there exist a finite number of Nash equilibria, and that at least one of these is nonrandomized. Finally, with the aid of simulation experiments, we explore the extent to which the Nash equilibria are characteristic of customer joining behaviour under a learning rule based on system-wide data.

scholarly journals On the Nash equilibria for the FCFS queueing system with load-increasing service rate

2005 ◽  
Vol 37 (02) ◽  
pp. 461-481 ◽  
Author(s):  
A. C. Brooms
Keyword(s):  
Queue Length ◽  
Nash Equilibria ◽  
Decision Rules ◽  
Learning Rule ◽  
Service Rate ◽  
Simulation Experiments ◽  
Player Game ◽  
Expected Sojourn Time ◽  
Increasing Function ◽  
Fcfs Discipline

We consider a service system (QS) that operates according to the first-come-first-served (FCFS) discipline, and in which the service rate is an increasing function of the queue length. Customers arrive sequentially at the system, and decide whether or not to join using decision rules based upon the queue length on arrival. Each customer is interested in selecting a rule that meets a certain optimality criterion with regard to their expected sojourn time in the system; as a consequence, the decision rules of other customers must be taken into account. Within a particular class of decision rules for an associated infinite-player game, the structure of the Nash equilibrium routeing policies is characterized. We prove that, within this class, there exist a finite number of Nash equilibria, and that at least one of these is nonrandomized. Finally, with the aid of simulation experiments, we explore the extent to which the Nash equilibria are characteristic of customer joining behaviour under a learning rule based on system-wide data.

scholarly journals Pathwise optimality of the exponential scheduling rule for wireless channels

2004 ◽  
Vol 36 (04) ◽  
pp. 1021-1045 ◽  
Author(s):  
Sanjay Shakkottai ◽  
R. Srikant ◽  
Alexander L. Stolyar
Keyword(s):  
Queue Length ◽  
Wireless Channels ◽  
Unique Feature ◽  
Heavy Traffic ◽  
Wireless Channel ◽  
Service Rate ◽  
Resource Pooling ◽  
Scheduling Policy ◽  
Multiple Data ◽  
Heavy Traffic Limit

We consider the problem of scheduling the transmissions of multiple data users (flows) sharing the same wireless channel (server). The unique feature of this problem is the fact that the capacity (service rate) of the channel varies randomly with time and asynchronously for different users. We study a scheduling policy called the exponential scheduling rule, which was introduced in an earlier paper. Given a system withNusers, and any set of positive numbers {an},n= 1, 2,…,N, we show that in a heavy-traffic limit, under a nonrestrictive ‘complete resource pooling’ condition, this algorithm has the property that, for each timet, it (asymptotically) minimizes maxnanq̃n(t), whereq̃n(t) is the queue length of usernin the heavy-traffic regime.

Queues with time-dependent arrival rates. III — A mild rush hour

1968 ◽  
Vol 5 (3) ◽  
pp. 591-606 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Queue Length ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Diffusion Approximations ◽  
Dimensionless Form ◽  
Queue Lengths ◽  
The Mean ◽  
Mean Queue Length

The arrival rate of customers to a service facility is assumed to have the form λ(t) = λ(0) — βt2 for some constant β. Diffusion approximations show that for λ(0) sufficiently close to the service rate μ, the mean queue length at time 0 is proportional to β–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for all λ(0) and β. Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

scholarly journals Nonzero-Sum Stochastic Games and Mean-Field Games with Impulse Controls

2021 ◽  
Author(s):  
Matteo Basei ◽  
Haoyang Cao ◽  
Xin Guo
Keyword(s):  
Nash Equilibria ◽  
Stochastic Games ◽  
Mean Field ◽  
Sufficient Conditions ◽  
Mean Field Games ◽  
Model Parameters ◽  
Management Problem ◽  
Player Game ◽  
The Impact ◽  
Impulse Controls

We consider a general class of nonzero-sum N-player stochastic games with impulse controls, where players control the underlying dynamics with discrete interventions. We adopt a verification approach and provide sufficient conditions for the Nash equilibria (NEs) of the game. We then consider the limiting situation when N goes to infinity, that is, a suitable mean-field game (MFG) with impulse controls. We show that under appropriate technical conditions, there exists a unique NE solution to the MFG, which is an ϵ-NE approximation to the N-player game, with [Formula: see text]. As an example, we analyze in detail a class of two-player stochastic games which extends the classical cash management problem to the game setting. In particular, we present numerical analysis for the cases of the single player, the two-player game, and the MFG, showing the impact of competition on the player’s optimal strategy, with sensitivity analysis of the model parameters.

scholarly journals Pathwise optimality of the exponential scheduling rule for wireless channels

2004 ◽  
Vol 36 (4) ◽  
pp. 1021-1045 ◽  
Author(s):  
Sanjay Shakkottai ◽  
R. Srikant ◽  
Alexander L. Stolyar
Keyword(s):  
Queue Length ◽  
Wireless Channels ◽  
Unique Feature ◽  
Heavy Traffic ◽  
Wireless Channel ◽  
Service Rate ◽  
Resource Pooling ◽  
Scheduling Policy ◽  
Multiple Data ◽  
Heavy Traffic Limit

We consider the problem of scheduling the transmissions of multiple data users (flows) sharing the same wireless channel (server). The unique feature of this problem is the fact that the capacity (service rate) of the channel varies randomly with time and asynchronously for different users. We study a scheduling policy called the exponential scheduling rule, which was introduced in an earlier paper. Given a system with N users, and any set of positive numbers {an}, n = 1, 2,…, N, we show that in a heavy-traffic limit, under a nonrestrictive ‘complete resource pooling’ condition, this algorithm has the property that, for each time t, it (asymptotically) minimizes maxnanq̃n(t), where q̃n(t) is the queue length of user n in the heavy-traffic regime.

scholarly journals On the Rate of Convergence and Limiting Characteristics for a Nonstationary Queueing Model

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 678 ◽  
Author(s):  
Yacov Satin ◽  
Alexander Zeifman ◽  
Anastasia Kryukova
Keyword(s):  
Rate Of Convergence ◽  
Queue Length ◽  
Arrival Rate ◽  
Service Rate ◽  
Queueing Model ◽  
Simple Method ◽  
Numerical Example

Consideration is given to the nonstationary analogue of M / M / 1 queueing model in which the service happens only in batches of size 2, with the arrival rate λ ( t ) and the service rate μ ( t ) . One proposes a new and simple method for the study of the queue-length process. The main probability characteristics of the queue-length process are computed. A numerical example is provided.

Optimal control of an M/G/1 queue with imperfectly observed queue length when the input source is finite

1991 ◽  
Vol 28 (01) ◽  
pp. 210-220
Author(s):  
Kazuyoshi Wakuta
Keyword(s):  
Optimal Control ◽  
Queue Length ◽  
Sufficient Conditions ◽  
Initial Time ◽  
Service Rate ◽  
Service Cost ◽  
Markov Decision ◽  
Input Source ◽  
Holding Cost ◽  
The Times

We consider the optimal control of an M/G/1 queue with finite input source. The queue length, however, can be only imperfectly observed through the observations at the initial time and the times of successive departures. At these times, the service rate can be chosen, based on the observable histories. A service cost and a holding cost are incurred. We show that such a control problem can be formulated as a semi-Markov decision process with imperfect state information, and present sufficient conditions for the existence of an optimal stationary I-policy.

ANALYSIS OF A BULK QUEUE WITH FAST AND SLOW SERVICE RATES AND MULTIPLE VACATIONS

2005 ◽  
Vol 22 (02) ◽  
pp. 239-260 ◽  
Author(s):  
R. ARUMUGANATHAN ◽  
K. S. RAMASWAMI
Keyword(s):  
Performance Measures ◽  
Queue Length ◽  
Queueing System ◽  
Cost Model ◽  
Probability Generating Function ◽  
Service Rate ◽  
Multiple Vacations ◽  
Bulk Queue ◽  
Service Rates ◽  
Number Of Customers

We analyze a Mx/G(a,b)/1 queueing system with fast and slow service rates and multiple vacations. The server does the service with a faster rate or a slower rate based on the queue length. At a service completion epoch (or) at a vacation completion epoch if the number of customers waiting in the queue is greater than or equal to N (N > b), then the service is rendered at a faster rate, otherwise with a slower service rate. After finishing a service, if the queue length is less than 'a' the server leaves for a vacation of random length. When he returns from the vacation, if the queue length is still less than 'a' he leaves for another vacation and so on until he finally finds atleast 'a' customers waiting for service. After a service (or) a vacation, if the server finds atleast 'a' customers waiting for service say ξ, then he serves a batch of min (ξ, b) customers, where b ≥ a. We derive the probability generating function of the queue size at an arbitrary time. Various performance measures are obtained. A cost model is discussed with a numerical solution.

A new heavy traffic limit for the asymmetric shortest queue problem

1999 ◽  
Vol 10 (5) ◽  
pp. 497-509 ◽  
Author(s):  
CHARLES KNESSL
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Parabolic Partial Differential Equation ◽  
Service Rate ◽  
Heavy Traffic Limit ◽  
Dimensional Approximation ◽  
Shortest Queue ◽  
Shortest Queue Problem

We consider the classic shortest queue problem in the heavy traffic limit. We assume that the second server works slowly and that the service rate of the first server is nearly equal to the arrival rate. Solving for the (asymptotic) joint steady state queue length distribution involves analyzing a backward parabolic partial differential equation, together with appropriate side conditions. We explicitly solve this problem. We thus obtain a two-dimensional approximation for the steady state queue length probabilities.

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