5.8 Threeserver queue: evaluating a new technology A
certain
circular highway is patrolled by three public safety cars. Each car
patrols a 1mile sector of the 3mile highway (see Figure P5.8). Calls
for assistance occur along the highway. A dispatcher assigns a car to
each call, if at least one is available. We wish to examine various
operating properties of this system.
The system operates as follows:
 Call positions are uniformly, independently distributed over the
circular highway.
 The call arrival process is a homogeneous Poisson process with
rate parameter calls per hour.
 Service time at the scene of the call has a negative exponential
distribution with mean ^{1} = ½
hour.
 Travel time is negligibly small compared to service time at the
scene.
 Speed of response is always 30 miles/hr.
 Uturns are permissible everywhere.
For parts (a)(c), assume that the dispatching strategy is as
follows. Given a call from sector i (i = 1, 2, 3):
 Assign car i, if available.
 Otherwise, randomly choose some car j (j i), and assign it, if at least one other
car is available.
 Otherwise, the call is lost.
 Find the steadystate probability that i cars are busy
(i = 0, 1, 2, 3).
 Find the steadystate probability that car 1 is busy and car 2 is
free.
 Find the average travel time to calls for this system. Evaluate
for 0, = 3, = 1,000.
 It has been proposed that the public safety bureau should
purchase a perfect resolution car locator system. With such a system,
the dispatching strategy is changed as follows:
Given a call from sector i (i = 1, 2, 3):
 Assign the closest available car, if at least one is
available;
 Otherwise, the call is lost.
Find the average travel time to a call for this system. Evaluate for
0, = 3, = 1,000. (This
part will utilize your knowledge of geometrical probability concepts.)
