**Strategically Seeking Service: How Competition Can Generate Poisson
Arrivals**

with Jan A. Van
Mieghem

## Winter 2004

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Abstract

We consider a simple timing game in which strategic agents select
arrival times to a service facility. Agents find congestion costly and hence
try to arrive when the system is under-utilized. Working in discrete time, we
characterize pure strategy equilibria for the case of ample service capacity.
In this case, agents try to spread themselves out as much as possible and their
self-interested actions will lead to a socially optimal outcome if all agents
have the same well-behaved delay cost function. For even modest sized problems,
the set of possible pure strategy equilibria is quite large, making
implementation potentially cumbersome. We consequently examine mixed strategy
equilibria and show that there is a unique symmetric equilibrium. Not only is
this equilibrium independent of the number of agents and their individual delay
cost functions, the arrival pattern it generates approaches a Poisson process
as the number of agents and arrival points gets large. Our results extend to
the case of limited capacity given an appropriate initialization of the system.
In this setting, we also argue that for any initialization, competition among
customers will equate the expected workload across the horizon and thus move
the system to steady state very quickly. Our model lends support to the
traditional literature on strategic behavior in queuing systems. This work has
generally assumed that customers arrive according to a renewal process but act
strategically upon arrival. We show that assuming renewal arrivals is an
acceptable assumption given a large population and long horizon.

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