Coupled queues with customer impatience

Motivated by assembly processes, we consider a Markovian queueing system with multiple coupled queues and customer impatience. Coupling means that departures from all constituent queues are synchronised and that service is interrupted whenever any of the queues is empty and only resumes when all queues are non-empty again. Even under Markovian assumptions, the state space grows exponentially with the number of queues involved. To cope with this inherent state space explosion problem, we investigate performance by means of two numerical approximation techniques based on series expansions, as well as by deriving the fluid limit. In addition, we provide closed-form expressions for the first terms in the series expansion of the mean queue content for the symmetric coupled queueing system. By an extensive set of numerical experiments, we show that the approximation methods complement each other, each one being accurate in a particular subset of the parameter space. (C) 2017 Elsevier B.V. All rights reserved

Performance analysis of hybrid MTS/MTO systems with stochastic demand and production

We present a comprehensive numerical approach with reasonably light complexity in terms of implementation and computation for assessing the performance of hybrid make-to-stock (MTS)/make-to-order (MTO) systems. In such hybrid systems, semi-finished products are produced up front and stored in a decoupling inventory. When an order arrives, the products are completed and possibly customised. We study this system in a stochastic setting: demand and production are modelled by random processes. In particular, our model includes two coupled Markovian queues: one queue represents the decoupling inventory and the other the order backlog. These queues are coupled as order processing can only occur when both queues are non-empty. We rely on matrix analytic techniques to study the performance of the MTO/MTS system under non-restrictive stochastic assumptions. In particular, we allow for arrival correlation and non-exponential setup and MTS and MTO processing times, while the hybrid MTS/MTO system is managed by an (s,S)-type threshold policy that governs switching from MTO to MTS and back. By some numerical examples, we assess the impact of inventory control, irregular order arrivals, setup and order processing times on inventory levels and lead times

Large deviations analysis for the M/H2/n+MM/H_2/n + M queue in the Halfin-Whitt regime

We consider the FCFS $M/H_2/n + M$ queue in the Halfin-Whitt heavy traffic regime. It is known that the normalized sequence of steady-state queue length distributions is tight and converges weakly to a limiting random variable W. However, those works only describe W implicitly as the invariant measure of a complicated diffusion. Although it was proven by Gamarnik and Stolyar that the tail of W is sub-Gaussian, the actual value of $\lim_{x \rightarrow \infty}x^{-2}\log(P(W >x))$ was left open. In subsequent work, Dai and He conjectured an explicit form for this exponent, which was insensitive to the higher moments of the service distribution. We explicitly compute the true large deviations exponent for W when the abandonment rate is less than the minimum service rate, the first such result for non-Markovian queues with abandonments. Interestingly, our results resolve the conjecture of Dai and He in the negative. Our main approach is to extend the stochastic comparison framework of Gamarnik and Goldberg to the setting of abandonments, requiring several novel and non-trivial contributions. Our approach sheds light on several novel ways to think about multi-server queues with abandonments in the Halfin-Whitt regime, which should hold in considerable generality and provide new tools for analyzing these systems