An Analysis of the Problem
Let ri be the long-run probability that there are i customers in the system. Then we have
the following equations:
| r0 |
= |
(1 - a)r0 + qr1 |
| r1 |
= |
ar0 + (1 - p - q)r1 + qr2 |
| r2 |
= |
pr1 + (1 - p - q)r2 + qr3 |
| : |
: |
: |
| ri |
= |
pri-1 + (1 - p - q)ri + qri+1 |
| : |
: |
: |
| rN-2 |
= |
prN-3 + (1 - p - q)rN-2 + qrN-1 |
| rN-1 |
= |
prN-2 + (1 - p - q)rN-1 + brN |
| rN |
= |
prN-1 + (1 - b)rN |
From the first equation, we have
From the second equation, we have
From the third equation, we have
Inductively we can show that
| ri = |
a
¾
q |
( |
p
¾
q |
)i-1r0 for i = 1, 2, ..., N - 1 |
and by the last equation
| rN = |
a
¾
b |
( |
p
¾
q |
)N-1r0 for i = 1, 2, ..., N - 1 |
Hence all the ri can be written in terms of a, b, p, q and
r0. It remains to find r0.
To determine r0, we note that
| 1 |
= |
r0 + r1 + ... + rN |
|
= |
| r0 |
æ
ç
è |
1 + |
a
¾
q |
+ |
a
¾
q |
( |
p
¾
q |
) + ... + |
a
¾
q |
( |
p
¾
q |
)N-2 + |
a
¾
b |
( |
p
¾
q |
)N-1 |
ö
÷
ø |
|
Therefore
| r0-1 = |
ì
ï
ï
í
ï
ï
î |
| 1 + |
a
¾¾
q-p |
(1 - ( |
p
¾
q |
) N-1 ) + |
a
¾
b |
( |
p
¾
q |
) N-1 if p ¹ q |
| |
| (N - 1) |
a
¾
q |
+ 1 + |
a
¾
b |
if p = q |
|
|
