On the assignment of customers to parallel queues

Advances in Applied Probability, 1987

We prove a monotonicity result for the problem of optimal service rate control in certain queueing networks. Consider, as an illustrative example, a number of ·/M/1 queues which are arranged in a cycle with some number of customers moving around the cycle. A holding cost hi (xi ) is charged for each unit of time that queue i contains xi customers, with hi being convex. As a function of the queue lengths the service rate at each queue i is to be chosen in the interval , where cost ci (μ) is charged for each unit of time that the service rate μis in effect at queue i. It is shown that the policy which minimizes the expected total discounted cost has a monotone structure: namely, that by moving one customer from queue i to the following queue, the optimal service rate in queue i is not increased and the optimal service rates elsewhere are not decreased. We prove a similar result for problems of optimal arrival rate and service rate control in general queueing networks. The results are ...

Probability in the Engineering and …, 1990

Theoretical Computer Science, 1997

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.

International Journal of Operations & …, 1997

Mathematics

In this paper, we study the problem of optimal routing for the pair of two-server heterogeneous queues operating in parallel and subsequent optimal allocation of customers between the servers in each queue. Heterogeneity implies different servers in terms of speed of service. An open-loop control assumes the static resource allocation when a router has no information about the state of the system. We discuss here the algorithm to calculate the optimal routing policy based on specially constructed Markov-modulated Poisson processes. As an alternative static policy, we consider an optimal Bernoulli splitting which prescribes the optimal allocation probabilities. Then, we show that the optimal allocation policy between the servers within each queue is of threshold type with threshold levels depending on the queue length and phase of an arrival process. This dependence can be neglected by using a heuristic threshold policy. A number of illustrative examples show interesting properties o...

Applied Stochastic Models in Business and Industry, 2010

Queueing networks with finite buffers, multiple servers, arbitrary topologies, and general service time distributions are considered in this paper. An approach to optimally allocate servers to series, merge, and split topologies and their combinations is demonstrated. The methodology builds upon two-moment approximations to the service time distribution embedded in the generalized expansion method for computing the performance measures in complex finite queueing networks and Powell's method for optimally allocating the servers within the network.

Journal of Communications and Networks

Article that has been accepted for inclusion in a future issue of a journal. Content is final as presented, with the exception of pagination.

IEEE Transactions on Automatic Control, 1990

Consider a set of k(22) heterogeneous and exponential servers which operate in parallel. Customers arrive into a single infinite capacity buffer according to a Poisson process, and are routed to available servers in accordance with some routing policy. We show that for arrival rates in some positive interval (0, XO], every routing policy which minimizes the long-run expected holding cost is contained in the set of routing policies that minimize the expected flow time for a system with fixed initial population and no new arrivals.

Queueing Systems, 2010

Consider a system of two queues in parallel, one of which is a •|M|1 single-server infinite capacity queue, and the other a •|G (N) |∞ batch service queue. A stream of general arrivals choose which queue to join, after observing the current state of the system, and so as to minimize their own expected delay. We show that a unique user equilibrium (user optimal policy) exists and that it possesses various monotonicity properties, using sample path and coupling arguments. This is a very simplified model of a transportation network with a choice of private and public modes of transport. Under probabilistic routing (which is equivalent to the assumption that users have knowledge only of the mean delays on routes), the network may exhibit the Downs-Thomson paradox observed in transportation networks with expected delay increasing as the capacity of the •|M|1 queue (private transport) is increased. We give examples where state-dependent routing mitigates the Downs-Thomson effect observed under probabilistic routing, and providing additional information on the state of the system to users reduces delay considerably. Keywords User equilibria • User optimal policies • Parallel queues • State dependent routing • Downs-Thomson paradox • Generalized threshold policies

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.

Institute of Mathematical Statistics Lecture Notes - Monograph Series, 2000

A continously operating system consisting of N Ki-out-oϊ-Ni subsystems connected in parallel is considered. The components of all subsystems are assumed identical with life times independent exponentially distributed random variables and the system is maintained by a single repaiman. Repair times are also assumed identical independent exponentials. We are interested in characterizing the allocation policy of the repairman which maximizes the system reliability at any time instant t (if any). In the present paper, we give a partial characterization of the optimal policy for systems consisting of highly reliable components using dynamic programming techniques. We also compute the leading term of a power series expansion of the reliability of the system at an arbitrary time instant t under the optimal policy. Finally, these results are extended to the problem of controlling the corresponding network of parallel queues in a scheduling problem with long mean arrival times and in its dual routing problem with long mean processing times.

Probability in the Engineering and …, 1992

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...

Discrete Event Dynamic Systems, 2006

In this paper we investigate the problem of the effective computation of the optimal routing sequence in a queuing system made of two queues with exponential service time in parallel. We first show that the optimal policy (minimizing the expected waiting time) is a Sturmian sequence and we establish several qualitative properties of this policy (monotonicity, continuity, convexity) Then, we propose an algorithm to compute the optimal routing sequence. We address the issues of time complexity as well as numerical stability of this algorithm. We then run an extensive set of experiments which show several interesting features of the optimal policy with apparent discontinuities and a fractal behavior.