Managing Queues with Heterogeneous Servers

Queueing Systems, 2004

An optimal policy to minimize the queue length in a multi-server controllable queueing system with heterogeneous servers has a threshold property, and it uses the fastest server if necessary (see and ). This study gives a numerical description of optimal policies that minimize the operational cost for such a system.

Theoretical Computer Science, 1997

Queueing Systems - Theory and Applications - QUESTA, 2002

This paper considers a heterogeneous M/G/2 queue. The service times at server 1 are exponentially distributed, and at server 2 they have a general distribution B(·). We present an exact analysis of the queue length and waiting time distribution in case B(·) has a rational Laplace–Stieltjes transform. When B(·) is regularly varying at infinity of index -?, we determine the tail behaviour of the waiting time distribution. This tail is shown to be semi-exponential if the arrival rate is lower than the service rate of the exponential server, and regularly varying at infinity of index 1-? if the arrival rate is higher than that service rate.

arXiv: Optimization and Control, 2020

Retailers use a variety of mechanisms to enable sales and delivery. A relatively new offering by companies is curbside pickup where customers purchase goods online, schedule a pickup time, and come to a pickup facility to receive their orders. To model this new service structure, we consider a queuing system where each arriving job has a preferred service completion time. Unlike most queuing systems, we make a strategic decision for when to serve each job based on their requested times and the associated costs. We assume that all jobs must be served before or on their requested time period, and the jobs are outsourced when the capacity is insufficient. Costs are incurred for jobs that are outsourced or served early. For small systems, we show that optimal capacity allocation policies are of threshold type. For general systems, we devise heuristic policies based on similar threshold structures. Our numerical study investigates the performance of the heuristics developed and shows the...

Naval Research Logistics (NRL), 2020

Recent studies reveal significant overdispersion and autocorrelation in arrival data at service systems such as call centers and hospital emergency departments. These findings stimulate the needs for more practical non-Poisson customer arrival models, and more importantly, new staffing formulas to account for the autocorrelative features in the arrival model. For this purpose, we study a multiserver queueing system where customer arrivals follow a doubly stochastic Poisson point process whose intensities are driven by a Cox-Ingersoll-Ross (CIR) process. The nonnegativity and autoregressive feature of the CIR process makes it a good candidate for modeling temporary dips and surges in arrivals. First, we devise an effective statistical procedure to calibrate our new arrival model to data which can be seen as a specification of the celebrated expectation-maximization algorithm. Second, we establish functional limit theorems for the CIR process, which in turn facilitate the derivation of functional limit theorems for our queueing model under suitable heavy-traffic regimes. Third, using the corresponding heavy traffic limits, we asymptotically solve an optimal staffing problem subject to delay-based constraints on the service levels. We find that, in order to achieve the designated service level, such an autoregressive feature in the arrival model translates into notable adjustment in the staffing formula, and such an adjustment can be fully characterized by the parameters of our new arrival model. In this respect, the staffing formulas acknowledge the presence of autoregressive structure in arrivals. Finally, we extend our analysis to queues having customer abandonment and conduct simulation experiments to provide engineering confirmations of our new staffing rules.

Communications in Computer and Information Science, 2016

Heterogeneous servers which can differ in service speed and reliability are becoming more popular in the modelling of modern communication systems. For a two-server queueing system with one nonreliable server and constant retrial discipline we formulate an optimal allocation problem for minimizing a long-run average cost per unit of time. Using a Markov decision process formulation we prove a number of monotone properties for the increments of the dynamic-programming value function. Such properties imply the optimality of a two-level threshold control policy. This policy prescribes the usage of a less productive server if the number of customers in the queue becomes higher than a predefined level which depends on the state of a non-reliable more powerful server. We provide also a heuristic solution for the optimal threshold levels in explicit form as a function of system parameters.

European Journal of Operational Research, 2010

We study M=M=c queues (c ¼ 1, 1 < c < 1 and c ¼ 1Þ in a 2-phase (fast and slow) Markovian random environment, with impatient customers. The system resides in the fast phase (phase 1) an exponentially distributed random time with parameter g and the arrival and service rates are k and l, respectively. The corresponding parameters for the slow phase (phase 0) are c, k 0 , and l 0 ð6 lÞ. When in the slow phase, customers become impatient. That is, each customer, upon arrival, activates an individual timer, exponentially distributed with parameter n. If the system does not change its environment from 0 to 1 before the customer's timer expires, the customer abandons the queue never to return.

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.

Mathematics, 2020

The paper studies a controllable multi-server heterogeneous queueing system where servers 1 operate at different service rates without preemption, i.e. the service times are uninterrupted. The 2 optimal control policy allocates the customers between the servers in such a way that the mean 3 number of customers in the system reaches its minimal value. The Markov decision model and the 4 policy-iteration algorithm are used to calculate the optimal allocation policy and corresponding mean 5 performance characteristics. The optimal policy, when neglecting the weak influence of slow servers, 6 is of threshold type defined as a sequence of threshold levels which specifies the queue lengths 7 for the usage of any slower server. To avoid time-consuming calculations for systems with a large 8 number of servers, we focus here on a heuristic evaluation of the optimal thresholds and compare this 9 solution with the real values. We develop also the simple lower and upper bound methods based on 10 approximation by an equivalent heterogeneous queueing system with a preemption to measure the 11 mean number of customers in the system operating under the optimal policy. Finally, the simulation 12 technique is used to provide sensitivity analysis of the heuristic solution to changes in the form of 13 inter-arrival and service time distributions. 14 Keywords: Heterogeneous servers; Markov decision process; policy-iteration algorithm; mean 15 number of customers; decomposable semi-regenerative process 16

International Series in Operations Research &amp; 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.

Manufacturing & Service Operations Management, 2009

Proof of Theorem 1: First consider the decision variable µ. We first prove the result for the single-server case (S = 1), and then deal with the general multi-server case. The proof is based on the sample path approach. Specifically, we prove that Y (viewed as a function of µ) satisfies sample path convexity (a term that has been introduced by Shaked and Shanthikumar (1988)). Specifically, let 0 ≤ µ 1 ≤ µ 2 ≤ µ 3 ≤ µ 4 be four service rates such that µ 1 + µ 4 = µ 2 + µ 3 , and fix λ, β(•) and η(•). Suppose that there exist Y 1 , ..., Y 4 , which are versions of the original head-count processes (Y i has service rate of µ i) that satisfy the following two properties for all t ≥ 0: 1. Y 1 (t) + Y 4 (t) ≥ Y 2 (t) + Y 3 (t), a.s. 2. Y 1 (t) ≥ max{Y 2 (t), Y 3 (t), Y 4 (t)}, a.s.

TOP, 2019

The problem of queues and waiting times is part of our daily life and so it is a situation that deserves a thorough study. Queueing theory mathematically studies the waiting lines and is part of the operations research field. This problem involves more complexity since it considers: the arrival process of the agents (customers) according to some probability distribution; the service time distribution and the number of available servers (in line or in parallel); and, finally, the queue discipline that determines the method used to serve the agents: first come, first served; last come, first served; etc. Chun presents a nice survey about the recent results on queueing problems where: there is only one server; the service time is the same for all agents (normalized to one); agents arrive according to some stochastic process; congestion may occur, and so the agents incur in waiting costs. The objective in this model, introduced by Dolan (1978), is to find an allocation rule that fixes the order in which the agents should be served and the monetary transfers. This problem can be addressed from several different approaches. We can assume an administrator is in charge of determining the order of the agents and the monetary transfers. But to do so, the administrator needs to know the waiting cost of each of the agents. We can assume that this is public information or we can assume it is private information and so an agent might reveal a different waiting cost if that is profitable for her. In the latter, it is important to provide incentives for the agents to reveal their true waiting costs. A queueing problem for a group of agents N is a vector θ = (θ i) i∈N , where θ i stands for the waiting cost of agent i. A solution, or a rule, for this problem is a pair (σ, t), where for each i, σ i denotes the position in the queue and t i denotes her monetary transfer. It is clear that to minimize the aggregated waiting cost, agents should be served according to their waiting cost in non-increasing order (queue efficiency). The property of queue efficiency means that the queue method applied in this problem is a priority queueing discipline that assigns a priority level, the waiting cost θ i in this case, to each customer and they are served following such priority on the first come first served basis.

Theoretical Computer Science, 1992

El-Taha, M. and S. Stidham Jr., Deterministic analysis of queueing systems with heterogeneous servers, Theoretical Computer Science 106 (1992) 243-264.

2011

Many-Server Queues with Time-Varying Arrivals, Customer A bandonment and Non-Exponential Distributions Yunan Liu This thesis develops deterministic heavy-traffic fluid appr oximations for many-server stochastic queueing models. The queueing models, with many homogenous servers working independently in parallel, are intended to model largescale service systems such as call centers and health care systems. Such models also have been e mploy d to study communication, computing and manufacturing systems. The heavytraffic approximations yield relatively simple formulas for quantities describing syst em performance, such as the expected number of customers waiting in the queue. The new performance approximations are valuable because, i n the generality considered, these complex systems are not amenable to exact mathem atical analysis. Since the approximate performance measures can be computed quite rap idly, they usefully complement more cumbersome computer simulation. Thus these heavy -t...

Systems & control letters, 1995