Exact Solutions for M/M/c/Setup Queues

Recently multiserver queues with setup times have been extensively studied because they have applications in power-saving data centers. The most challenging model is the M/M/$c$/Setup queue where a server is turned off when it is idle and is turned on if there are some waiting jobs. Recently, Gandhi et al.~(SIGMETRICS 2013, QUESTA 2014) present the recursive renewal reward approach as a new mathematical tool to analyze the model. In this paper, we derive exact solutions for the same model using two alternative methodologies: generating function approach and matrix analytic method. The former yields several theoretical insights into the systems while the latter provides an exact recursive algorithm to calculate the joint stationary distribution and then some performance measures so as to give new application insights.Comment: Submitted for revie

A batch-service queueing model with a discrete batch Markovian arrival process

Queueing systems with batch service have been investigated extensively during the past decades. However, nearly all the studied models share the common feature that an uncorrelated arrival process is considered, which is unrealistic in several real-life situations. In this paper, we study a discrete-time queueing model, with a server that only initiates service when the amount of customers in system (system content) reaches or exceeds a threshold. Correlation is taken into account by assuming a discrete batch Markovian arrival process (D-BMAP), i.e. the distribution of the number of customer arrivals per slot depends on a background state which is determined by a first-order Markov chain. We deduce the probability generating function of the system content at random slot marks and we examine the influence of correlation in the arrival process on the behavior of the system. We show that correlation merely has a small impact on the threshold that minimizes the mean system content. In addition, we demonstrate that correlation might have a significant influence on the system content and therefore has to be included in the model

Order batching in multi-server pick-and-sort warehouses.

In many warehouses, customer orders are batched to profit from a reduction in the order picking effort. This reduction has to be offset against an increase in sorting effort. This paper studies the impact of the order batching policy on average customer order throughput time, in warehouses where the picking and sorting functions are executed separately by either a single operator or multiple parallel operators. We present a throughput time estimation model based on Whitt's queuing network approach, assuming that the number of order lines per customer order follows a discrete probability distribution and that the warehouse uses a random storage strategy. We show that the model is adequate in approximating the optimal pick batch size, minimizing average customer order throughput time. Next, we use the model to explore the different factors influencing optimal batch size, the optimal allocation of workers to picking and sorting, and the impact of different order picking strategies such as sort-while-pick (SWP) versus pick-and-sort (PAS)Order batching; Order picking and sorting; Queueing; Warehousing;

Performance of the Gittins Policy in the G/G/1 and G/G/k, With and Without Setup Times

How should we schedule jobs to minimize mean queue length? In the preemptive M/G/1 queue, we know the optimal policy is the Gittins policy, which uses any available information about jobs' remaining service times to dynamically prioritize jobs. For models more complex than the M/G/1, optimal scheduling is generally intractable. This leads us to ask: beyond the M/G/1, does Gittins still perform well? Recent results indicate that Gittins performs well in the M/G/k, meaning that its additive suboptimality gap is bounded by an expression which is negligible in heavy traffic. But allowing multiple servers is just one way to extend the M/G/1, and most other extensions remain open. Does Gittins still perform well with non-Poisson arrival processes? Or if servers require setup times when transitioning from idle to busy? In this paper, we give the first analysis of the Gittins policy that can handle any combination of (a) multiple servers, (b) non-Poisson arrivals, and (c) setup times. Our results thus cover the G/G/1 and G/G/k, with and without setup times, bounding Gittins's suboptimality gap in each case. Each of (a), (b), and (c) adds a term to our bound, but all the terms are negligible in heavy traffic, thus implying Gittins's heavy-traffic optimality in all the systems we consider. Another consequence of our results is that Gittins is optimal in the M/G/1 with setup times at all loads.Comment: 41 page

Delay analysis of a two-class batch-service queue with class-dependent variable server capacity

In this paper, we analyse the delay of a random customer in a two-class batch-service queueing model with variable server capacity, where all customers are accommodated in a common single-server first-come-first-served queue. The server can only process customers that belong to the same class, so that the size of a batch is determined by the length of a sequence of same-class customers. This type of batch server can be found in telecommunications systems and production environments. We first determine the steady state partial probability generating function of the queue occupancy at customer arrival epochs. Using a spectral decomposition technique, we obtain the steady state probability generating function of the delay of a random customer. We also show that the distribution of the delay of a random customer corresponds to a phase-type distribution. Finally, some numerical examples are given that provide further insight in the impact of asymmetry and variance in the arrival process on the number of customers in the system and the delay of a random customer

An all geometric discrete-time multiserver queueing system

In this work we look at a discrete-time multiserver queueing system where the number of available servers is distributed according to one of two geometrics. The arrival process is assumed to be general independent, the service times deterministically equal to one slot and the buffer capacity infinite. The queueing system resides in one of two states and the number of available servers follows a geometric distribution with parameter determined by the system state. At the end of a slot there is a fixed probability that the system evolves from one state to the other, with this probability depending on the current system state only, resulting in geometrically distributed sojourn times. We obtain the probability generating function (pgf) of the system content of an arbitrary slot in steady-state, as well as the pgf of the system content at the beginning of an arbitrary slot with a given state. Furthermore we obtain an approximation of the distribution of the delay a customer experiences in the proposed queueing system. This approximation is validated by simulation and the results are illustrated with a numerical example

Exploring a new Markov chain model for multiqueue systems.

Traditionally, Markov models have been used to study multiserver systems using exhaustive or gated service. In addition, exhaustive-limited and gate-limited models have also been used in communication systems to reduce overall latency. Recently the authors have proposed a new Markov Chain approach to study gate-limited service. Multiqueue systems such as polling systems, in which the server serves various queues have also been extensively studied but as a separate branch of queueing theory. This paper proposes to describe multiqueue systems in terms of a new Markov Chain called the Zero-Server Markov Chain (ZSMC). The model is used to derive a formula for the waiting times in an exhaustive polling system. An intuitive result is obtained and this is used to develop an appoximate method which works well over normal operational ranges