Dinard van der Laan

Periodic routing to parallel queues and billiard sequences

Abstract.In this companion paper of [10] we introduce the combinatorial notion of unbalance for a routing pattern. Using this unbalance we derive an upper bound for the total average expected waiting time of jobs which are routed to parallel queues according to a periodic routing rule. A billiard sequence is obtained with unbalance smaller than or equal to −1, where N is the number of different symbols in the sequence which corresponds to the number of parallel queues in the routing problem.

Routing Jobs to Servers with Deterministic Service Times

In this paper we consider the problem of routing deterministic arriving jobs to parallel servers with deterministic (distinct) service times, where we assume that the arrival rate of the jobs is equal to the total service capacity of the servers. Our goal is to find routing policies that minimize the long-run average waiting time of the arriving jobs. We give lower and upper bounds for the minimal long-run average waiting time, and we present results on the structure of optimal policies. We derive mathematical programming problems that can be solved to obtain optimal routing policies, and for a single queue we have results on routing according to a regular sequence to that queue. Finally, we discuss several algorithms to obtain within reasonable time good but generally not optimal deterministic routing policies.

On The Optimal Open-Loop Control Policy For Deterministic And Exponential Polling Systems

In this article, we consider deterministic (both fluid and discrete) polling systems with N queues with infinite buffers and we show how to compute the best polling sequence (minimizing the average total workload). With two queues, we show that the best polling sequence is always periodic when the system is stable and forms a regular sequence. The fraction of time spent by the server in the first queue is highly noncontinuous in the parameters of the system (arrival rate and service rate) and shows a fractal behavior. Moreover, convexity properties are shown and are used in a generalization of the computation of the optimal control policy (in open loop) for the stochastic exponential case.

On the Average Waiting Time for Regular Routing to Deterministic Queues

We consider a deterministic queueing system in whichN = 2 servers of different speeds operate in parallel. Each service in queuei takes the deterministic timeS i. Identical customers arrive exactly one per time unit, and it is desirable to route them to the queues so that the average waiting time (we consider as waiting time the time a customer waits in the buffer of a queue, and thus the service time is not included in this) is minimized. We provide an algorithm to compute lower and upper bounds on this quantity. The upper bound is found by showing that there is a periodic policy for which the average waiting time is no greater than the lower bound plus ( N / 2) - 1. Thus, the bounds coincide whenN = 2. For obtaining the lower bound, we give explicit formulae for the average waiting time in case of regular routing to a deterministic queue.

Optimal Balanced Control for Call Centers

In this paper we study the optimal assignment of tasks to agents in a call center. For this type of problem, typically, no single deterministic and stationary (i.e., state independent and easily implementable) policy yields the optimal control, and mixed strategies are used. Other than finding the optimal mixed strategy, we propose to optimize the performance over the set of “balanced” deterministic periodic non-stationary policies. We provide a stochastic approximation algorithm that allows to find the optimal balanced policy by means of Monte Carlo simulation. As illustrated by numerical examples, the optimal balanced policy outperforms the optimal mixed strategy.

NOTE ON THE CONVEXITY OF THE STATIONARY WAITING TIME AS A FUNCTION OF THE DENSITY

It is proved that the total average waiting time is a convex function of the routing densities for regular routing to parallel queues.

The unbalance and bounds on the average waiting time for periodic routing to one queue

Abstract.In this paper we introduce the combinatorial notion of unbalance for a periodic zero-one splitting sequence. Using this unbalance we derive an upper bound for the average expected waiting time of jobs which are routed to one queue according to a periodic zero-one splitting sequence. In the companion paper [16] the upper bound will be extended to the routing to N parallel queues.

Approximate Results for a Generalized Secretary Problem

A version of the classical secretary problem is studied, in which one is interested in selecting one of the b best out of a group of n differently ranked persons who are presented one by one in a random order. It is assumed that b is a preassigned (natural) number. It is known, already for a long time, that for the optimal policy one needs to compute b position thresholds, for instance via backwards induction. In this paper we study approximate policies, that use just a single or a double position threshold, albeit in conjunction with a level rank. We give exact and asymptotic (as n tends to infinity) results, which show that the double-level policy is an extremely accurate approximation.

Assigning multiple job types to parallel specialized servers

In this paper methods of mixing decision rules are investigated and applied to the so-called multiple job type assignment problem with specialized servers. This problem is modeled as continuous time Markov decision process. For this assignment problem performance optimization is in general considered to be difficult. Moreover, for optimal dynamic Markov decision policies the corresponding decision rules have in general a complicated structure not facilitating a smooth implementation. On the other hand optimization over the subclass of so-called static policies is known to be tractable. In the current paper a suitable static decision rule is mixed with dynamic decision rules which are selected such that these rules are relatively easy to describe and implement. Some mixing methods are discussed and optimization is performed over corresponding classes of so-called mixing policies. These mixing policies maintain the property that they are easy to describe and implement compared to overall optimal dynamic Markov decision policies. Besides for all investigated instances the optimized mixing policies perform substantially better than optimal static policies.