A queueing system with on-demand servers: local stability of fluid limits

Abstract

We study a system where a random flow of customers is served by servers (called agents) invited on-demand. Each invited agent arrives into the system after a random time; after each service completion, an agent returns to the system or leaves it with some fixed probabilities. Customers and/or agents may be impatient, that is, while waiting in queue, they leave the system at a certain rate (which may be zero). We consider the queue-length-based feedback scheme, which controls the number of pending agent invitations, depending on the customer and agent queue lengths and their changes. The basic objective is to minimize both customer and agent waiting times. We establish the system process fluid limits in the asymptotic regime where the customer arrival rate goes to infinity. We use the machinery of switched linear systems and common quadratic Lyapunov functions to approach the stability of fluid limits at the desired equilibrium point and derive a variety of sufficient local stability conditions. For our model, we conjecture that local stability is in fact sufficient for global stability of fluid limits; the validity of this conjecture is supported by numerical and simulation experiments. When local stability conditions do hold, simulations show good overall performance of the scheme.

Appendix: Some results for switched linear systems and CQLF

The following facts are used in the proof of our main result (Theorem 2).

Proposition 1

([16]) Let $L(\lambda ) = \det (A - \lambda I) = 0$ be the characteristic equation of a matrix A in $\mathbb {R}^{3 \times 3}$:

$$\begin{aligned} L(\lambda ) = a_0 \lambda ^3 + a_1 \lambda ^2 + a_2 \lambda + a_3 = 0 \ , \ a_0 > 0. \end{aligned}$$

(66)

Matrix A is Hurwitz if and only if $a_1$, $a_2$, $a_3$ are positive and $a_1 a_2 > a_0 a_3$.

Proposition 2

([10, 19]) The existence of a CQLF for the LTI systems is sufficient for the exponential stability of the switched linear system.

Proposition 3

([10, 19]) Let $A_1$ and $A_2$ be Hurwitz matrices in $\mathbb {R}^{n \times n}$, and the difference $A_1 - A_2$ have rank one. Then, two systems $u^\prime (t) = A_1 u(t)$ and $u^\prime (t) = A_2 u(t)$ have a CQLF if and only if the matrix product $A_1 A_2$ has no negative real eigenvalues.

Proposition 4

([18]) If $A_1^{-1}$ is non-singular, the product $A_1 A_2$ has no negative eigenvalues if and only if $A_1^{-1} + \tau A_2$ is non-singular for all $\tau \ge 0$.