Analysis of the adaptive MMAP[K]/PH[K]/1 queue: A multi-type queue with adaptive arrivals and general impatience

In this paper we introduce the adaptive MMAP[K] arrival process and analyze the adaptive MMAP[K]/PH[K]/1 queue. In such a queueing system, customers of K different types with Markovian inter-arrival times and possibly correlated customer types, are fed to a single server queue that makes use of r thresholds. Service times are phase-type and depend on the type of customer in service. Type k customers are accepted with some probability a(l,k) if the current workload is between threshold i - 1 and i. The manner in which the arrival process changes its state after generating a type k customer also depends on whether the customer is accepted or rejected. The solution method exists in reducing the joint workload and arrival process to a fluid queue with r thresholds, the steady state of which is expressed using matrix analytic methods. The time and memory complexity of this approach is also shown to be linear in the number of thresholds, allowing us to study systems with thousands of thresholds. Markovian multi-type queues with customer impatience form a subclass of the queues considered in this paper. A numerical method to determine the probability of abandonment and the waiting time distribution is provided if the patience distributions have finite support, while for general customer impatience numerical examples show that accurate approximate results can be obtained using a step-function approach. Numerical examples with adaptive sources that model certain types of admission and congestion control are also included. (C) 2012 Elsevier B.V. All rights reserved.