Karl Sigman

Analysis of join-the-shortest-queue routing for web server farms

Join the Shortest Queue (JSQ) is a popular routing policy for server farms. However, until now all analysis of JSQ has been limited to First-Come-First-Serve (FCFS) server farms, whereas it is known that web server farms are better modeled as Processor Sharing (PS) server farms. We provide the first approximate analysis of JSQ in the PS server farm model for general job-size distributions, obtaining the distribution of queue length at each queue. To do this, we approximate the queue length of each queue in the server farm by a one-dimensional Markov chain, in a novel fashion. We also discover some interesting insensitivity properties of PS server farms with JSQ routing, and discuss the near-optimality of JSQ.

Sampling at subexponential times, with queueing applications

We study the tail asymptotics of the r.v. X(T) where {X(t)} is a stochastic process with a linear drift and satisfying some regularity conditions like a central limit theorem and a large deviations principle, and T is an independent r.v. with a subexponential distribution. We find that the tail of X(T) is sensitive to whether or not T has a heavier or lighter tail than a Weibull distribution with tail . This leads to two distinct cases, heavy tailed and moderately heavy tailed, but also some results for the classical light-tailed case are given. The results are applied via distributional Littles law to establish tail asymptotics for steady-state queue length in GI/GI/1 queues with subexponential service times. Further applications are given for queues with vacations, and M/G/1 busy periods.

A Pollaczek–Khintchine formula for M / G /1 queues with disasters

A disaster occurs in a queue when a negative arrival causes all the work (and therefore customers) to leave the system instantaneously. Recent papers have addressed several issues pertaining to queueing networks with negative arrivals under the i.i.d. exponential service times assumption. Here we relax this assumption and derive a Pollaczek–Khintchine-like formula for M / G /1 queues with disasters by making use of the preemptive LIFO discipline. As a byproduct, the stationary distribution of the remaining service time process is obtained for queues operating under this discipline. Finally, as an application, we obtain the Laplace transform of the stationary remaining service time of the customer in service for unstable preemptive LIFO M / G /1 queues.

Asymptotic convergence of scheduling policies with respect to slowdown

We explore the performance of an M/GI/1 queue under various scheduling policies from the perspective of a new metric: the slowdown experienced by the largest jobs. We consider scheduling policies that bias against large jobs, towards large jobs, and those that are fair, e.g., processor-sharing (PS). We prove that as job size increases to infinity, all work conserving policies converge almost surely with respect to this metric to no more than 1/(1 - ρ), where ρ denotes the load. We also find that the expected slowdown under any work conserving policy can be made arbitrarily close to that under PS, for all job sizes that are sufficiently large.

The stability of open queueing networks

The stability of open Jackson networks is established where service times are i.i.d. general distribution, exogeneous interarrival times are i.i.d. general distribution, and the routing is Markovian. The service time distributions are only required to have finite first moment. The system is modeled (at arrival epochs) as a general state space Markov chain. Explicit regeneration points are found (even in the case when the system never empties) and the chain is shown to be Harris ergodic if standard rate conditions are enforced, that is, if at each node, the long run average amount of work per unit time that arrives exogenously destined for that node is strictly less than one. In addition, we prove that if the system is modeled in continuous time then convergence to a steady-state occurs in total variation if the interarrival time distribution is spread-out. Extensions of the results to multi-server nodes, non-Markovian routing and Markov modulated arrivals are given.

A review of regenerative processes

The authors present an expository review of the theory of regenerative processes starting with the more traditional notions and then moving on to some of the more recent and modern developments. Fi...

Light traffic heuristic for an M/G/ 1 queue with limited inventory

Motivated by solving a stylized location problem, we develop a light traffic heuristic for anM/G/1 queue with limited inventory that gives rise to a closed form expression for average delay in terms of basic system parameters. Simulation experiments show that the heuristic works well. The inventory operates as follows: the inventory level drops by one unit after each service completion and whenever it drops to a pre-specified levelu, an order is placed with replenishment time ∼ exp(γ). Upon replenishment the inventory is restocked to a pre-specified levels and any arrivals when there is no inventory are placed in queue. Suggestions are given to cover the more general case of a New Better than Used (NBU) replenishment time distribution. Applications to inventory management problems are also discussed.

Delay moments for FIFO GI/GI/s queues

For stable FIFO GI/GI/s queues, s ≥ 2, we show that finite (k+1)st moment of service time, S, is not in general necessary for finite kth moment of steady-state customer delay, D, thus weakening some classical conditions of Kiefer and Wolfowitz (1956). Further, we demonstrate that the conditions required for E[Dk]<∞ are closely related to the magnitude of traffic intensity ρ (defined to be the ratio of the expected service time to the expected interarrival time). In particular, if ρ is less than the integer part of s/2, then E[D] < ∞ if E[S3/2]<∞, and E[Dk]<∞ if E[Sk]<∞, k≥ 2. On the other hand, if s-1 < ρ < s, then E[Dk]<∞ if and only if E[Sk+1]<∞, k ≥ 1. Our method of proof involves three key elements: a novel recursion for delay which reduces the problem to that of a reflected random walk with dependent increments, a new theorem for proving the existence of finite moments of the steady-state distribution of reflected random walks with stationary increments, and use of the classic Kiefer and Wolfowitz conditions.

Notes on the stability of closed queueing networks

A new proof of the stability of closed Jackson-type queueing networks (with general service-time distributions) is given and sufficient conditions are given for obtaining Cesaro, weak and total variation convergence of the continuous-time joint queue length and residual service-time process to a limiting distribution. The result weakens the sufficient conditions (for stability) of Borovkov (1986) by allowing more general service-time distributions.

Work-modulated queues with applications to storage processes

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 n th customers service time, S n , depends upon their delay, D n , 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 Fosters 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.