Bounds for performance characteristics : a systematic approach via cost structures

In this paper we present a systematic approach to the construction of bounds for the average costs in Markov chains with possibly infinitely many states. The technique used to prove the bounds is based on dynamic programming. Most performance characteristics of Markovian systems can be represented by the average costs for some appropriately chosen cost structure. Therefore, the approach can be used to generate bounds for relevant performance characteristics. The approach is demonstrated for the shortest queue model. It is shown how for this model several bounds for the mean waiting time can be constructed. We include numerical results to demonstrate the quality of these bound

Shortest Expected Delay Routing for Erlang Servers

The queueing problem with a Poisson arrival stream and two identical Erlang servers is analysed for the queueing discipline based on shortest expected delay. This queueing problem may be represented as a random walk on the integer grid in the first quadrant of the plane. In the paper it is shown that the equilibrium distribution of this random walk can be written as a countable linear combination of product forms. This linear combination is constructed in a compensation procedure. In this case the compensation procedure is essentially more complicated than in other cases where the same idea was exploited. The reason for the complications is that in this case the boundary consists of several layers which in turn is caused by the fact that transitions starting in inner states are not restricted to end in neighbouring states. Good starting solutions for the compensation procedure are found by solving the shortest expected delay problem with the same service distributions but with instantaneous jockeying. It is also shown that the results can be used for an efficient computation of relevant performance criteria

On the Finite Capacity Shortest Queue Problem

We consider two parallel queues. There is one server tending to each queue and the capacity of each queue is K. The network is fed by a single Poisson arrival stream of rate λ, and the two servers are identical exponential servers working at rate µ. A new arrival is routed to the queue with the smaller number of customers. If both have the same number of customers then the arrival is routed randomly, with the probability of joining either queue being 1/2. If there are more than 2K customers in the system, further arrivals are turned away and lost. We let ρ = λ/µ and take K →∞, and consider the cases ρ 2 and ρ − 2 = O(K−1). We shall obtain asymptotic approximations to the joint steady state distribution of finding m customers in the first queue and n in the second. The asymptotic approximations are shown to be quite accurate numerically. We shall identify precisely for what ranges of m and n can the finite capacity model be approximated by the infinite capacity one. We will also show that the marginal distribution of finding n customers in the second queue undergoes a transition when ρ = 4. Key words: Shortest queue problem; Finite capacity; Poisson arrival stream; Analytical approximation

Randomized load balancing in finite regimes

Randomized load balancing is a cost efficient policy for job scheduling in parallel server queueing systems whereby, with every incoming job, a central dispatcher randomly polls some servers and selects the one with the smallest queue. By exactly deriving the jobs' delay distribution in such systems, in explicit and closed form, Mitzenmacher~\cite{Mi03} proved the so-called `power-of-two' result, which states that by randomly polling only two servers yields an exponential improvement in delay over randomly selecting a single server. Such a fundamental result, however, was obtained in an asymptotic regime in the total number of servers, and does do not necessarily provide accurate estimates for practical finite regimes with small or moderate number of servers. In this paper we obtain stochastic lower and upper bounds on the jobs' average delay in non-asymptotic/finite regimes, by borrowing ideas for analyzing the particular case of Join-the-Shortest-Queue (JSQ) policy. Numerical illustrations indicate not only that the obtained (lower) bounds are remarkably accurate, but also that the existing exact but asymptotic results can be largely misleading in finite regimes (e.g., by more than $100\%$ in the case of $12$ servers)