Generation of random variates for PHDs, MAPs and BMAPs -

Abstract

This research proposes the simulation of three arrival processes: phase-type (PH) process-distribution, Markovian arrival process (MAP) and batch Markovian arrival process (BMAP). Two simulation models are developed and utilized to randomly generate inter-arrival times, i.e. the time headway between two successive event occurrences. PHDs, MAPs and BMAPs do not belong to the distributions or stochastic processes that are commonly used in simulation tools, but it is usually straightforward to integrate them into simulation software by drawing on the underlying Markov chains which govern the activity of these processes. Building stochastic simulation models based on the underlying Markov chain becomes extensive and error-prone for processes with higher orders, lagging in both time and traceability. Therefore, an alternative approach to simulating these processes is proposed, such that only the start and end states of an arrival epoch, rather than its whole transition activity, are utilized to set a cumulative distribution for the inter-arrival time. Since the inter-arrival time understudy is a matrix exponential, then the corresponding cumulative distribution function cannot be inverted and the classical inverse transform method cannot be applied. In this context, discretization of the function is an appropriate alternative whereby a database of inter-arrival times and their corresponding cumulative probabilities is formulated and then randomly sampled to generate inter-event times. The scope of work comprises of conducting: (1) the simulation of the underlying Markov chain such that arrivals and their corresponding arrival times are recorded and alternatively (2) the discretization of the cumulative distribution function indicated by the start and end states of arrival epochs and random sampling of the latter to produce random inter-arrival times. Approaches (1) and (2) are applied on several examples and compared in terms of accuracy and efficiency. Results suggest that the approach (2) is capable of performin