Work-modulated queues with applications to storage processes

Abstract

We study two FIFO single-server queueing models in which both the arrival and service processes are modulated by the amount of work in the system. In the first model, the nth customer's service time, Sn, depends upon their delay, Dn, in a general Markovian way and the arrival process is a non-stationary Poisson process (NSPP) modulated by work, that is, with an intensity that is a general deterministic function g of work in system V(t). Some examples are provided. In our second model, the arrivals once again form a work-modulated NSPP, but, each customer brings a job consisting of an amount of work to be processed that is i.i.d. and the service rate is a general deterministic function r of work. This model can be viewed as a storage (dam) model (Brockwell et al. (1982)), but, unlike previous related literature, (where the input is assumed work-independent and stationary), we allow a work-modulated NSPP. Our approach involves an elementary use of Foster's criterion (via Tweedie (1976)) and in addition to obtaining new results, we obtain new and simplified proofs of stability for some known models. Using further criteria of Tweedie, we establish sufficient conditions for the steady-state distribution of customer delay and sojourn time to have finite moments.

On level crossings for a general class of piecewise-deterministic Markov processes

“…Processes covered by Scenario 4.3 appear in queueing theory and also as storage processes. For instance, we can consider a work-modulated single-server queue, (X t ), being the workload process (for a formal definition and detailed treatment of such a process, we refer the reader to the discussion of 'Model 2' in [9]; see also [27]). Then we will have µ(y) ≤ 0, y > 0, and µ(0) = 0.…”

“…Special cases have been extensively studied in the literature. We just mention storage processes [14], [27], stress release models [7], [30], [31], queueing models [9], [27], and repairable systems [21]. It is mostly assumed that J (x, ·) does not depend on x ∈ R and that the jumps are either only nonnegative or only nonpositive.…”

A make-to-stock mountain-type inventory model

“…We assume that the conditions for stability are ful…lled so that Y = lim t!1 Y (t) represents the equilibrium random variable of the mountain (the latter limit is de…ned in terms of weak convergence). We refer to [7] for an extensive discussion of stability conditions for queueing models with work-modulated arrival and/or service times; one may translate the increments during 0-periods in our model into service requirements that depend on the work found upon arrival, and subsequently use the stability conditions from Model 1 of [7].…”

“…With these values we …nd (d 1 ; d 2 ) = (5:236; 0:764), and (C 1 ; C 2 ) = (0:276; 0:724) from which we can calculate the c.d.f. G(t) given in (7). To solve the ODE (10) numerically for f 0 (x), we impose the boundary conditions…”

Finite-Buffer Queues with Workload-Dependent Service and Arrival Rates

“…In the sequel, it is assumed that λ(·), r(·), B(·) are chosen such that the steady-state distribution of the infinite-buffer version, that is, for g(w, b, K) = w + b, exists (and then for all g(·, ·, ·)). For details on stability and existence of steady-state distributions, we refer to [8,9].…”