On the optimal assignment of customers to parallel servers

Abstract

We consider a queuing system with several identical servers, each with its own queue. Identical customers arrive according to some stochastic process and as each customer arrives it must be assigned to some server's queue. No jockeying amongst the queues is allowed. We are interested in assigning the arriving customers so as to maximize the number of customers which complete their service by a certain time. If each customer's service time is a random variable with a non-decreasing hazard rate then the strategy which does this is one which assigns each arrival to the shortest queue.

Contrôle dynamique de flux dans un système d'attente avec panne

“…En supposant que la distribution de temps de service est la même pour les deux files, Winston [15], Weber [12], Ephremides, Varaiya et Walrand [2] ont montré qu'envoyer les clients dans la file la moins chargée est une politique fortement optimale.…”

Distributed routing for load balancing

“…If this were possible each routing unit would at each time send an arriving task to the server with the shortest queue in case µ 1 =µ 2 . This minimizes the expected waiting time in the class of all policies which assign a task upon its arrival to a server [20,45,46]. Of course one can do even better if one postpones the assignment decision until a server becomes free, or equivalently if unlimited jockeying between the queues is possible (See [4] for a discussion on how much further improvement in performance this can give).…”

“…Again when all the routing units have the same information at all times and hence can be coordinated, then it does not matter that there is more than one arrival stream. Optimal control results such as "join the shortest queue" (JSQ) [45,46) can be used. Several extensions of this closed-loop optimal routing have been studied for more complicated systems [14,15,26,31,34).…”

Shortest expected delay routing for Erlang servers

“…We will do so for the situation of shortest expected delay rQuting which means that the jobs of all customers are considered as consisting of r exponentially distributed subjobs and a new job is joining the queue with the lowest number of subjobs to be executed (in case the numbers of subjobs are equal, either queue is joined with probabilitỹ ). Hordijk and Koole [13] and Weber [16] have shown that, within certain constraints, this way of routing is optimal. For us, however, the important aspect is that the model can be represented by a two-dimensional random walk in the first quadrant of the plane: Define i and j as the numbers of subjobs in both queues and define the state (m, n) by m = min(i,j), n = Ii -jl.…”