5.8 Three-server queue: evaluating a new technology A certain circular highway is patrolled by three public safety cars. Each car patrols a 1-mile sector of the 3-mile 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:

1. Call positions are uniformly, independently distributed over the circular highway.

2. The call arrival process is a homogeneous Poisson process with rate parameter calls per hour.

3. Service time at the scene of the call has a negative exponential distribution with mean ^-1 = ½ hour.

4. Travel time is negligibly small compared to service time at the scene.

5. Speed of response is always 30 miles/hr.

6. U-turns 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):

1. Assign car i, if available.

2. Otherwise, randomly choose some car j (j i), and assign it, if at least one other car is available.

3. Otherwise, the call is lost.

   1. Find the steady-state probability that i cars are busy (i = 0, 1, 2, 3).

   2. Find the steady-state probability that car 1 is busy and car 2 is free.

   3. Find the average travel time to calls for this system. Evaluate for 0, = 3, = 1,000.

   4. 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):

      1. Assign the closest available car, if at least one is available;

      2. 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.)