Mixed Polling with Rerouting and Applications

Queueing systems with a single server in which customers wait to be served at a finite number of distinct locations (buffers/queues) are called discrete polling systems. Polling systems in which arrivals of users occur anywhere in a continuum are called continuous polling systems. Often one encounters a combination of the two systems: the users can either arrive in a continuum or wait in a finite set (i.e. wait at a finite number of queues). We call these systems mixed polling systems. Also, in some applications, customers are rerouted to a new location (for another service) after their service is completed. In this work, we study mixed polling systems with rerouting. We obtain their steady state performance by discretization using the known pseudo conservation laws of discrete polling systems. Their stationary expected workload is obtained as a limit of the stationary expected workload of a discrete system. The main tools for our analysis are: a) the fixed point analysis of infinite dimensional operators and; b) the convergence of Riemann sums to an integral. We analyze two applications using our results on mixed polling systems and discuss the optimal system design. We consider a local area network, in which a moving ferry facilitates communication (data transfer) using a wireless link. We also consider a distributed waste collection system and derive the optimal collection point. In both examples, the service requests can arrive anywhere in a subset of the two dimensional plane. Namely, some users arrive in a continuous set while others wait for their service in a finite set. The only polling systems that can model these applications are mixed systems with rerouting as introduced in this manuscript.Comment: to appear in Performance Evaluatio

Random fixed points, systemic risk and resilience of heterogeneous financial network

We consider a large random network, in which the performance of a node depends upon that of its neighbours and some external random influence factors. This results in random vector valued fixed-point (FP) equations in large dimensional spaces, and our aim is to study their almost-sure solutions. An underlying directed random graph defines the connections between various components of the FP equations. Existence of an edge between nodes $i,j$ implies the $i$-th FP equation depends on the $j$-th component. We consider a special case where any component of the FP equation depends upon an appropriate aggregate of that of the random `neighbour' components. We obtain finite-dimensional limit FP equations in a much smaller dimensional space, whose solutions aid to approximate the solution of FP equations for almost all realizations, as the number of nodes increases. We use Maximum theorem for non-compact sets to prove this convergence. We apply the results to study systemic risk in an example financial network with large number of heterogeneous entities. We utilized the simplified limit system to analyse trends of default probability (probability that an entity fails to clear its liabilities) and expected surplus (expected-revenue after clearing liabilities) with varying degrees of interconnections between two diverse groups. We illustrated the accuracy of the approximation using exhaustive Monte-Carlo simulations. Our approach can be utilized for a variety of financial networks (and others); the developed methodology provides approximate small-dimensional solutions to large-dimensional FP equations that represent the clearing vectors in case of financial networks.Comment: 51 pages, 7 figures and 4 tables, will appear in Annals of Operations Researc

Fixed-point equations solving Risk-sensitive MDP with constraint

There are no computationally feasible algorithms that provide solutions to the finite horizon Risk-sensitive Constrained Markov Decision Process (Risk-CMDP) problem, even for problems with moderate horizon. With an aim to design the same, we derive a fixed-point equation such that the optimal policy of Risk-CMDP is also a solution. We further provide two optimization problems equivalent to the Risk-CMDP. These formulations are instrumental in designing a global algorithm that converges to the optimal policy. The proposed algorithm is based on random restarts and a local improvement step, where the local improvement step utilizes the solution of the derived fixed-point equation; random restarts ensure global optimization. We also provide numerical examples to illustrate the feasibility of our algorithm for inventory control problem with risk-sensitive cost and constraint. The complexity of the algorithm grows only linearly with the time-horizon.Comment: 8 pages, 4 figures, submitted to the 2023 American Control Conference (ACC

New results in Branching processes using Stochastic Approximation

We consider various types of continuous-time two-type population size-dependent Markov Branching Process (BP). The offspring distribution can depend on the current population (those alive at the given time) and or on the total population (dead and alive) of the two types. Using stochastic approximation techniques, we propose an ODE-based framework to study a general class of such BPs. We primarily focus on time-asymptotic proportion of the two types, via ODE limits; while, the ODE solution approximates certain normalized trajectories. In addition to extending the analysis of several existing BPs, we analyze two new variants: BP with attack and acquisition, and BP with proportion-dependent offsprings. Using these results, we study competition in viral markets and fake news control on online social networks.Comment: 54 pages; 1 table; 5 figure