Short run dynamics of multi-class queues (original) (raw)
To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems
International Series in Operations Research & Management Science, 2003
Preface xi 1. INTRODUCTION A non-cooperative game is defined as follows. Let N = {1,. .. , n} be a finite set of players and let A i denote a set of actions available to player i ∈ N. A pure strategy for player i is an action from A i. A mixed strategy corresponds to a probability function which prescribes a randomized rule for selecting an action from A i. Denote by S i the set of strategies available to player i. A strategy profile s = (s 1 ,. .. , s n) assigns a strategy s i ∈ S i to each player i ∈ N. Each player is associated with a real payoff function F i (s). This function specifies the payoff received by player i given that the strategy profile s is adopted by the players. Denote by s −i a profile for the set of players N \ {i}. The function F i (s) = F i (s i , s −i) is assumed to be linear in s i. This means that if s i is a mixture with 1 In case of periodicity, with period d, replace the limit by averaging the limits along d consecutive periods. Note that ∞ s=0 πs(δ) does not necessarily sum up to 1. On one hand, it can be greater than 1 (in fact, can even be unbounded) when more than one recurrent chain exists, and on the other hand it may sum up to 0. An example for the latter case is when λ > µ and δ(s) = join for all s ≥ 0. x F (x, y). We are interested in cases where x(y) is continuous and strictly monotone. Figure 1.1 illustrates a situation where a strategy corresponds to a nonnegative number. It depicts one instance where x(y) is monotone decreasing and another where it is monotone increasing. We call these situations avoid the crowd (ATC) and follow the crowd (FTC), respectively. The rationale behind this terminology is that in an FTC (respectively, ATC) case, the higher the values selected by the others, the higher (respectively, lower) is one's best response. 3 An interesting generalization to this rule is proposed by Balachandran and Radhakrishnan [19]. Suppose that waiting t time units costs Ce at for given parameters C > 0 and a ≥ 0. Then, the expected waiting cost of a customer is ∞ 0 Ce at w(t) dt where w(t) is the density function of the waiting time. In an M/M/1 system w(t) = (µ − λ)e −(µ−λ)t where λ is the arrival rate and µ is the service rate. In this case the expected cost equals C µ−a−λ. Note that the case of linear waiting costs is obtained when a = 0. 4 See Deacon and Sonstelie [43] and Png and Reitman [140] for empirical studies concerning this parameter. Examples for disciplines that are strong and work-conserving are FCFS, LCFS, random order, order which is based on customers payments, and EPS. Service requirements are assumed to be independent and identically distributed. Denote by µ −1 the (common) expected service requirement (i.e., µ is the rate of service). For stability, assume that the system's utilization factor ρ = λ µ is strictly less than 1 (sometimes, when individual optimization leads to stability, this assumption is removed). The following five results hold when the arrival process is Poisson with rate λ, the service distribution is exponential (an M/M/1 model) with rate µ, and the service discipline is strong and work-conserving. They also hold for M/G/1 models when the service discipline is either EPS or LCFS-PR. The probability that n (n ≥ 0) customers are in the system (at arbitrary times as well as at arrival times) is (1 − ρ)ρ n. (1.2) 11 When 3 5λ > 1, commuters appear at a rate so low that even when all of them use the shuttle service, the individual's best response is still to use the bus service. In other words, when λ < 3 5 , using the bus service is a dominant strategy. Chapter 2 OBSERVABLE QUEUES This chapter deals with queueing systems, where an arriving customer observes the length of the queue before making his decisions.
European Conference on Queueing Theory 2016
HAL (Le Centre pour la Communication Scientifique Directe), 2016
Kleinrock (1964) proposed a queueing discipline for a single-server queue in which customers from different classes accumulate priority as linear functions of their waiting time. When the server becomes free, it selects the waiting customer with the highest amount of accumulated priority at that instant, provided that the queue is nonempty. For such a queue, Kleinrock developed a recursion for calculating the expected waiting time of customers from each class. More recently, Stanford, Taylor and Ziedins (2014) took another look at this queue, which they termed the Accumulating Priority Queue (APQ), and derived the waiting time distributions for each class. Kleinrock and Finkelstein (1967) also studied an accumulating priority system in which customers' priorities increase as a power-law function of their time in the queue. They established that it is possible to associate a particular linear accumulating priority queue with such a power-law accumulating priority queue, in such a way that the expected waiting times of customers from the different classes are preserved. In this paper, we extend their analysis to characterise the class of nonlinear accumulating priority queues for which an equivalent linear APQ can be found, in the sense that the waiting time distributions for each of the classes are identical in both the linear and nonlinear systems.
Stability of Parallel Queueing Systems with Coupled Service Rates
Discrete Event Dynamic Systems, 2008
This paper considers a parallel system of queues fed by independent arrival streams, where the service rate of each queue depends on the number of customers in all of the queues. Necessary and sufficient conditions for the stability of the system are derived, based on stochastic monotonicity and marginal drift properties of multiclass birth and death processes. These conditions yield a sharp characterization of stability for systems, where the service rate of each queue is decreasing in the number of customers in other queues, and has uniform limits as the queue lengths tend to infinity. The results are illustrated with applications where the stability region may be nonconvex.
A Numerical Approach to Stability of Multi-class Queueing Networks
IEEE Transactions on Automatic Control, 2017
The Multi-class Queueing Network (McQN) arises as a natural multi-class extension of the traditional (single-class) Jackson network. In a single-class network subcriticality (i.e. subunitary nominal workload at every station) entails stability, but this is no longer sufficient when jobs/customers of different classes (i.e. with different service requirements and/or routing scheme) visit the same server; therefore, analytical conditions for stability of McQNs are lacking, in general. In this note we design a numerical (simulation-based) method for determining the stability region of a McQN, in terms of arrival rate(s). Our method exploits certain (stochastic) monotonicity properties enjoyed by the associated Markovian queue-configuration process. Stochastic monotonicity is a quite common feature of queueing models and can be easily established in the single-class framework (Jackson networks); recently, also for a wide class of McQNs, including first-come-first-serve (FCFS) networks, monotonicity properties have been established. Here, we provide a minimal set of conditions under which the method performs correctly. Eventually, we illustrate the use of our numerical method by presenting a set of numerical experiments, covering both single and multi-class networks.
There are three topics in the thesis. In the first topic, we addressed a control problem for a queueing system, known as the "N-system", under the Halfin-Whitt heavy traffic regime and a static priority policy was proposed and is shown to be asymptotically optimal, using weak convergence techniques. In the second topic, we focused on the hospitals, where faster servers(nurses), though work more efficiently, have the heavier workload, and the Randomized Most-Idle (RMI) routing policy was proposed to tackle this unfairness issue, trying to reward faster servers who serve more with less workload. we extended the existing result to show that this desirable property of the RMI policy holds under a system with multiple customer classes using theoretical exact analysis as well as numerical simulations. In the third topic, the problem was to decide an appropriate number of representatives over time according to the prescribed service quality level in the call center. We examined the stability of two methods which were designed to generate appropriate staffing functions on a simulated data and real call center data from an actual bank.
Stability and performance for multi-class queueing networks with infinite virtual queues
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On the stability of queueing networks and fluid models
The stability of the Lu-Kumar queueing network is re-analyzed. It is shown that the associated fluid network is a hybrid dynamical system that has a succession of invariant subspaces leading to global stability. It is explained why large enough stochastic perturbations of the production rates lead to an unstable queuing network while smaller perturbations do not change the stability. The two reasons for the instability are the breaking of the invariance of the subspaces and a positive Lyapunov exponent. A service rule that stabilizes the system is proposed.
Population effects in Multiclass Processor Sharing Queues
2009
Consider a single server queueing system with several classes of customers, each having its own renewal input process and its own general service times distribution. Upon completing service, customers may leave, or reenter the queue, possibly as customers of a different class. The server is operating under the egalitarian or the discriminatory processor sharing discipline. In this paper, we consider the fluid approximation of this multiclass processor sharing queue. We first provide the results allowing to compute the trajectories for this model, under the egalitarian PS discipline. Asymptotic results for overloaded queues are also stated. Next, we show that a simple transformation allows to compute the solution for the discriminatory PS queue as well. Finally, we illustrate the different results through numerical experiments. We compare transient trajectories with simulations, and we discuss the fairness issue that may arise in overloaded PS queues.
Analyzing an M|M| N Queueing System with Feedback by the Method of Asymptotic Analysis
Cybernetics and Systems Analysis, 2021
In the paper, we consider a mathematical model for repeated customers in the form of a queuing system with N servers, instant and delayed feedback, and an orbit. It is believed that the orbit size for repeated customers is infinite. The input flow is Poisson. To find the joint probability distribution of the number of occupied servers in the system and the number of customers in the orbit, the asymptotic analysis method is used. The results of a numerical experiment are presented.
Heavy-Traffic Analysis of a Non-Preemptive Multi-Class Queue with Relative Priorities
Probability in the Engineering and Informational Sciences, 2015
We study the steady-state queue-length vector in a multi-class queue with relative priorities. Upon service completion, the probability that the next served customer is from class k is controlled by class-dependent weights. Once a customer has started service, it is served without interruption until completion. We establish a state-space collapse for the scaled queue-length vector in the heavy-traffic regime, that is, in the limit the scaled queue-length vector is distributed as the product of an exponentially distributed random variable and a deterministic vector. We observe that the scaled queue length reduces as classes with smaller mean service requirement obtain relatively larger weights. We finally show that the scaled waiting time of a class-k customer is distributed as the product of two exponentially distributed random variables.