Dynamic Admission Game into an M/M/1 Queue

Abstract

Around 50 years ago, P. Naor has derived the optimal social and individual admission rules to an M/M/1 queue. In both cases, the optimal policies were identified to be of a pure threshold type: admit if and only if the number queued upon arrival is below some threshold. The value of the threshold in the individual optimal case was shown to be larger than the one for the social optimal criterion. We make the observation that admitting according to a threshold policy requires only the information of whether the queue is above or below a threshold. We call these “red” and “green” light, respectively, associated with a threshold, say L. The question that we pose in this paper is: what happens if one restricts to the above information pattern but let the threshold level L be chosen by the system which signals to arrivals whether the queue is above or below the threshold. Can one find a choice of a threshold that will induce an equilibrium that performs better than in the case that full information is available? We also examine the question of what is the threshold that maximizes the revenue for the queue. We show that the choice of threshold that maximizes the system’s performance at equilibrium is the same as under the full information case if the service in the queue follows the FIFO discipline.

Appendix: Uniform f-Geometric Ergodicity and the Continuity of the Markov Chain

The continuity of the steady state probabilities and thus of the expected queue length hold for the case of finite k since the chain is ergodic with finitely many states. We thus focus below on the case of infinite k. We show continuity of the expected queue length in q for $q_{{R}}$ restricted to some closed interval for which the corresponding value of $\underline{\rho }_{\,_{ R}}$ is smaller than 1. (Due to Lemma 1 there exists indeed an interval such that any policy for which $q_{{R}}$ is not in the interval cannot be optimal.)

We show that the Markov chain is f-Geometric Ergodic and then use Lemma 5.1 from Spieksma (1990).

Consider the Markov chain embedded at each transition in the queue size. Thus for $I \geq \max (L,1)$, with probability β the event is a departure and otherwise it is an arrival, where

$$\displaystyle{\beta:= \frac{\mu } {\mu +\nu + q_{{R}}\zeta }.}$$

Note α > 0 implies that β > 1∕2 (α is defined in the proof of Theorem 1).

Define $f(i) =\exp (\gamma i)$, for any $I \geq \max (L,1)$,

$$\displaystyle\begin{array}{rcl} & E[f(I_{t+1}) - f(I_{t})\vert I_{t} = i] =\beta \exp [\gamma (i - 1)] + (1-\beta )\exp [\gamma (i + 1)] -\exp (\gamma i)& {}\\ & = f(i)\varDelta \quad \mbox{ where}\quad \varDelta =\beta z^{-1} + (1-\beta )z - 1 & {}\\ \end{array}$$

and where $z:=\exp (-\gamma )$. Note that Δ = 0 at

$$\displaystyle{z_{1,2} = \frac{1 \pm \sqrt{1 - 4\beta (1-\beta )}} {2(1-\beta )} =\{ 1, \frac{\beta } {1-\beta }\}}$$

Thus Δ < 0 for all γ in the interval $\left (0,\log \left ( \frac{\beta }{1-\beta }\right )\right )$ (which is nonempty since we showed that 1 > β > 1∕2). We conclude that for any γ in that interval, f is a Lyapunov function and the Markov chain is γ-geometrically ergodic uniformly in q.