# The Asymmetric Inclusion Process: Tandem-Queue Networks with Unlimited Batch Service

### Tel Aviv University

### Résumé:

Inspired by the biological process of gene translation and its modeling as a flow of particles through a tandem array of unit-capacity ‘servers’ - known in biophysics as the Totally Asymmetric Exclusion Process (TASEP), we devise and analyze a new model: the Totally Asymmetric Inclusion Process (TASIP) [1]. The TASIP (or ASIP) is a system of n Markovian queues in tandem, each with unbounded batch service. That is, when service is completed at queue k, all particles present at that queue move simultaneously to queue k+1. Equations for the multi-dimensional time-dependent Probability Generating Function (PGF) of the system-state (occupancy) vector at arbitrary time instants, as well as for the PGF at ‘event epochs’ (arrival or service completion), are derived. We show that, at steady state, the two PGFs coincide – implying a PASTA-type phenomenon. We then explicitly derive the PGFs for systems with n=2, or with n=3, tandem queues, and construct an iterative procedure to derive the PGF for any larger number of queues. We also explicitly compute the mean of the system-state vector at steady state, and establish a Little-type law. Furthermore, we explicitly compute the PGF of the TASIP’s total load in steady state, and obtain a stochastic decomposition result, leading to a product-form solution. In addition, we show that the total load variable is independent of the order of the queues. We then show that TASIP models with identical service rates are, under various perspectives, optimal.
If time permits we will consider the stochastic asymptotic behavior of the ASIP model in three different limiting regimes: (i) heavy-traffic limit – in which the particles’ arrival rate tends to infinity; (ii) large-system limit – in which the system size (i.e., the number of tandem queues) tends to infinity; and (iii) balanced-system limit – in which both the system size and the service rate of each queue tend to infinity, while keeping the ratio between the two fixed.
Joint work with Shlomi Reuveni and Iddo Eliazar