Question

Consider a single server system with a limit of 3 jobs (an M/M/1/3 system). Let λ be the mean arrival rate and μ be the mean service rate.

(a) Use the singleton subset partition method to derive a system of balance equations (note the last equation is the probability norming equation):

λp0−μp1 =0 λp0+μp2−(λ+μ)p1 =0 λp1+μp3−(λ+μ)p2 =0 λp2−μp3 =0

p0+p1+p2+p3 =1.

(b) Use the subset partition between successive nodes to derive a system of balance equations.

(c) Solve for each pi in terms of p0 for each set of balance equations (a and b) to establish that they yield the same solution.

Answer 1

Solution(a):

Since this system can have at most 3 jobs, there are 3+1 possible states, , representing the number of jobs in the system. Interest is in developing the steady-state distribution of the number of jobs in the system. Assuming that a steady-state exists, then the flow into and out of each state must balance. Let denote the steady-state probability of n jobs in the system for .

Thus, the steady-state flow-balance equation for an intermediate state n is

,.............................................(1)

The two special flow-balance equations (for states 0 and 3) are

..........................................................(2)

The normalization condition is

Solution (b) Subset partition by separating each node into its own singleton subset which gives the following system of equations

   .....................................................(3)

The system (3) is the same as system (1) and (2).


Solution (c)

From equation (2), we get

Now putting in equation (1), we have

  

From equation (2), we get
   

Again from the normalization condition, we obtain

  

Since the system given in (a) and (b) are same,  they give the same solution.