Heavy-traffic and State-Space Collapse Sample Clauses

Heavy-traffic and State-Space Collapse. As we consider systems under heavy-traffic, we need to define several terms relevant in this setting. Definition 1 A random variable Y has an (α, β) impulse-exponential distribution if P (Y > 0) = α and P (Y > x|Y > 0) = e−βx for x > 0, for some 0 ≤ α ≤ 1 and β > 0. A stochastic process Z is said to have an (α, β) impulse-exponential steady-state if it has a well defined stationary distribution Z(∞) that is (α, β) impulse-exponential. Consider a parametric family of queueing systems, indexed by n, with each system fitting the description above where primitives are now superscripted by n. Thus ρn is the traffic intensity of the nth system. We assume that the base system described in the previous section is associated with some index n∗ so that ρn∗ = ρ < 1. If limn→∞ ρn = 1 then the sequence of systems is said to be going into heavy-traffic. Heavy- traffic limit theorems consider the limiting processes of such a sequence of systems. We denote by “ˆ” the limit of any system primitives; therefore pˆi = limn→∞ pn, i = 1, 2 and ρˆ = 1, by definition. If limn→∞ ρn = 1 then queue lengths will be exploding and must be scaled for the limit to provide any sort of meaningful results. In particular, if X˜n = Ψ(X, n) is some scaling of the number of customers in the system, then it is the limit of these scaled processes that are studied. This scaled limit process may then be unscaled to provide an approximation for system behavior in moderate traffic (i.e., Xn ≈ Ψ—1(Xˆ, n), where Xˆ is the limit of the processes X˜n). The convergence described in the previous paragraphs is weak convergence of stochastic processes and is usually defined either in terms of the Skorohod metric or in terms of uniform convergence on compact sets (see, e.g., Xxxxx 1990). Here, we merely assume that weak convergence, denoted by ⇒, is well defined given the sequence of systems under consideration. We prove our results under any heavy-traffic limit that results in an appropriate limit for scaled queue length. To wit, consider the following definition. Definition 2 Consider a sequence of systems fitting the assumptions above, operated under some policy π, and indexed by n such that ρn → 1. The number of customers in the system is scaled according to some deterministic function Ψ(·, n) and let the scaled total queue length be represented by Q˜n. We say that policy π, together with this scaling and primitives, has an (α, β) impulse-exponential heavy-traffic limit if Q˜n converges weakly to a stocha...