Tandem stochastic systems: Jackson networks, asymmetric exclusion processes, asymmetric inclusion processes and Catalan numbers

A tandem stochastic system is a network of n sites (queues) in series, where particles (customers, jobs, packets, etc.) move unidirectionally from one site to the next until they leave the system. When each site is an M/M/1 queue, i.e., where the buffer size of each site is unlimited and only single particles move between sites, the system is known as Tandem Jackson Network (TJN) [3]. The TJN is famous for its product-form solution of the multi-dimensional distribution function of the sites' queue-sizes (occupancies). Another well-known tandem stochastic system is the Asymmetric Simple Exclusion Process (ASEP) [1], where each site can hold at most a single particle, a constraint that causes blockings on particles' forward movements. The ASEP is a paradigmatic model in non-equilibrium statistical mechanics. In contrast, the newly introduced (Reuveni, Eliazar and Yechiali) Asymmetric Inclusion Process (ASIP) is a tandem series of sites, each with unbounded buffer capacity and with unlimited-size batch service [5]. That is, when service is completed at site k, all particles present there move simultaneously to site k + 1 and form a cluster together with the cluster of particles (if any) already residing in site k + 1. The ASIP is a showcase of complexity [6]. We analyze the innovative ASIP and show that its multi-dimensional Probability Generating Function (PGF) does not admit a product-form solution. Nevertheless, we present a method to calculate this PGF [5]. Surprisingly though, the load (total number of particles up to site k) does admit a product-form solution. It is consequently shown that homogeneous systems (i.e. systems with identical servers) are optimal for various load-related objective functions [5]. Considering the total number of particles in a site-interval of length m (m = 1, 2,..., n) that starts at site k, the corresponding probability generating functions create a discrete two-dimensional boundary-value problem which is solved explicitly [8]. Catalan's numbers, and their generalizations, arise naturally in this context [4]. It is further proved [8] that the probability of site k being positively occupied is proportional to 1/√k, while the variance in the occupancy of site k is proportional to √k. Finally, we derive limit laws (when the number of sites becomes large) for various system's variables [7]. In particular, we show that the 'load', as well as the 'draining time', each obeys a Gaussian distribution (with corresponding coefficients), while the 'inter-exit time' follows a Rayleigh distribution. An extension of the basic ASIP model with a fairly general arrival scheme, where gate opening intervals follow a Markov renewal process, is studied in [2]. The steady-state distribution of the total number of customers in the first k queues is determined.