The price of forgetting in parallel and non-observable queues

Abstract

We consider a broker-based network of non-observable parallel queues and analyze the minimum expected response time and the optimal routing policy when the broker has the memory of its previous routing decisions. We provide lower bounds on the minimum response time by means of convex programming that are tight, as follows by a numerical comparison with a proposed routing scheme. The Price of Forgetting (PoF), the ratio between the minimum response times achieved by a probabilistic broker and a broker with memory, is shown to be unbounded or arbitrarily close to one depending on the coe cient of variation of the service time distributions. In the case of exponential service times, the PoF is bounded from above by two, which is tight in heavy-tra c, and independent of the network size and heterogeneity. These properties yield a simple engineering product-form approximating tightly the minimum response time. Finally, we put our results in the context of game theory revisiting the Price of Anarchy (PoA) of parallel queues: It can be decomposed into the product of the PoA achieved by a probabilistic broker (already well understood) and the PoF. IntroductionIn the context of *-computing and manufacturing systems, users submit jobs without knowing which machine will handle their execution. A central broker is usually in charge of distributing incoming jobs to a set of resources to optimize some utility or cost function. The mean response time (simply response time in the following), the expected time it takes for a job to join the system and return to its issuer, is an important metric that is usually taken into account to achieve a better exploitation of resources and improve the average quality of service.The problem of nding the routing strategy (or policy) that minimizes response time and the resulting response time itself are well-known problem in queueing theory and the literature is overwhelming. There are two main classes of routing strategies: Closed-loop policies, where the broker knows the state of each resource (number of jobs), and open-loop policies, where the broker does not know their state. Computing the best closed-loop policy is known to be a hard problem and a lot of work has been devoted to the computation of good approximations; see, e.g., [9,27, 33] and the references therein.The focus of this paper is on the open-loop case, and more precisely on o -line policies. Indeed, the state of the queues may not to be known on-line to the broker because of a number of reasons whose e ects become worse and worse as the network size increases: i) Communicating the system state to the broker increases the network load, ii) the information received by the broker can be out of date, and iii) the synchronization that results from knowing the system state can degrade the performance [26]. A celebrated case is when the broker only knows the service time distributions of the network resources (or queues in the following) and dispatches jobs to queues according to an i.i.d. probabilistic law. The probabili...

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