Abstract
Performance measures are studied for a generalized n-site asymmetric simple inclusion process (G-ASIP), where a general process controls intervals between gate-opening instants. General formulae are obtained for the Laplace–Stieltjes transform, as well as the means, of the (i) traversal time, (ii) busy period, and (iii) draining time. The PGF and mean of (iv) the system’s overall load are calculated, as well as the probability of an empty system, along with (v) the probability that the first occupied site is site k (k = 1, 2, …, n). Explicit results are derived for the wide family of gamma-distributed gate inter-opening intervals (which span the range between the exponential and the deterministic probability distributions), as well as for the uniform distribution. It is further shown that a homogeneous system, where at gate-opening instants gate j opens with probability pj=1n, is optimal with regard to (i) minimizing mean traversal time, (ii) minimizing the system’s load, (iii) maximizing the probability of an empty system, (iv) minimizing the mean draining time, and (v) minimizing the load variance. Furthermore, results for these performance measures are derived for a homogeneous G-ASIP in the asymptotic cases of (i) heavy traffic, (ii) large systems, and (iii) balanced systems.
Highlights
A tandem stochastic system (TSS) is an n-site network in a series, where random events cause particles to propagate unidirectionally, under specific stochastic rules, from one site to the along the onedimensional array of sites until exiting the system
This paper extends the research on generalized asymmetric simple inclusion process (ASIP) systems where a general renewal process controls intervals between gate-opening instants
Expressions are derived for the Laplace–Stieltjes transform, or probability generating function (PGF), as well as the means of various performance measures
Summary
A tandem stochastic system (TSS) is an n-site network in a series, where random events cause particles (customers, messages, products, calls, jobs, molecules, etc.) to propagate unidirectionally, under specific stochastic rules, from one site to the along the onedimensional array of sites (queues, servers, stations, etc.) until exiting the system. The TJN is famous for its product-form solution of the joint multidimensional probability generating function (PGF) of the sites’ queue sizes (occupancies) Another notable TSS is the asymmetric simple exclusion process (ASEP) (see, e.g., [8,9,10,11]), where the buffer size of each site is only one, allowing sites to hold at most a single particle at a time. The current paper concentrates on deriving expressions for key performance measures for G-ASIP systems with the particles’ arrival at the first site and a general distribution of gate inter-opening times. We further show that in the intriguing case of balanced systems, the five measures have similar behavior and share the same limiting distributions when gate inter-opening times are: (i) exponential, (ii) deterministic, or (iii) uniform
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