Rate control under heavy traffic with strategic servers

Abstract

We consider a large queueing system that consists of many strategic servers that are weakly interacting. Each server processes jobs from its unique critically loaded buffer and controls the rate of arrivals and departures associated with its queue to minimize its expected cost. The rates and the cost functions in addition to depending on the control action, can depend, in a symmetric fashion, on the size of the individual queue and the empirical measure of the states of all queues in the system. In order to determine an approximate Nash equilibrium for this finite player game we construct a Lasry-Lions type mean-field game (MFG) for certain reflected diffusions that governs the limiting behavior. Under conditions, we establish the convergence of the Nash-equilibrium value for the finite size queuing system to the value of the MFG.

A Numerical Scheme for a Mean Field Game in Some Queueing Systems Based on Markov Chain Approximation Method

“…Abusing notation, when Assumption 2.4 holds, we will write b(t, x, u) = b(t, η 0 , x, u), y(t) = y(t, η 0 ), and r(t) = r(t, η 0 ). The following is one of the main results from [5].…”

“…The following conditions were used in [5] in order to characterize the value function V ν and the optimal control. As argued in [5], Berge's maximum theorem (see [4,Theorem 17.31]) together with part (b) of the above assumption implies thatα is continuous. Also note that (2.5) implies that b, f, g, y, r are bounded functions, in particular,…”

“…For further discussion about the assumption, see [5]. The Hamilton-Jacobi-Bellman equation for the value function V ν (t, x) is given as follows.…”

“…heavy traffic limit) and the number of queues approach ∞, simultaneously. It is shown in [5] that given a solution of the MFG and an optimal control associated with it of the form in Section 2.1, one can construct an asymptotic (in number of players and in heavy traffic limit) Nash equilibrium for the n-player game such that the solution of the MFG and its associated value function approximate the empirical distribution of the states of the queues and the value function of each server. The current work provides a numerical approximation for the solution of the MFG that is needed in order for constructing the above n-player asymptotic Nash equilibrium.…”

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We use the Markov chain approximation method to construct approximations for the solution of the mean field game (MFG) with reflecting barriers studied in [5]. The MFG is formulated in terms of a controlled reflected diffusion with a cost function that depends on the reflection terms in addition to the standard variables: state, control, and the mean field term.

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