Birth and Death Processes
Example: Communication System
• A communication system has two processers that decode messages and a buffer that
holds at most two messages.
• A message that arrives when the buffer is full is lost.
• Each processer needs an average of 2 minutes to decode a message.
• Messages arrive at a rate of 1 per minute.
Assume that inter-arrival times and processing times are independent exponential random
variables.
(a) Draw a state diagram with possible states and corresponding birth/death rates.
Solution:
1
1
1
0
.5
1
1
2
3
1
4
1
1
Let T1 = time until processer 1 finishes and T2 = time until processer 2 finishes.
E[Ti ] = 2 i = 1, 2 implies that rate parameter for both distributions are 1/2 i.e.,
T1 , T2 ∼ iid Exp(1/2)
For k = 1, implying that µ1 = 21 = .5
For k ≥ 2, D = min{T1 , T2 } giving D ∼ Exp( 12 +
1
2
= 1); thus µk = 1,
(b) What is the (large t) probability that the system is empty?
S =1+
implying that p0 =
1
S
=
1
1·1
1·1·1
1·1·1·1
+
+
+
=9
.5 1 · .5 1 · 1 · .5 1 · 1 · 1 · .5
1
9
(c) What is the pmf of X?
pk = P [X = k] =
k
pk = P (X = k)
1
λ0 λ1 . . . λk−1
p0
µ1 µ2 . . . µk
0
1
2
3
4
1
9
2
9
2
9
2
9
2
9
k = 2, 3, 4
(d) What is the (large t) expected # of jobs in the system?
E[X] = 0 · p0 + 1 · p1 + 2 · p2 + 3 · p3 + 4 · p4
1 2
= 0 · + · (1 + 2 + 3 + 4)
9 9
= 20/9
(d) What is the probability that an incoming in message is turned away?
An incoming message is lost when system is full i.e., X = 4
P (message lost) = P (system full)
= P (X = 4)
= p4 = 2/9
(e) What is the expected number of messages in the buffer?
Let Y = # of messages in the buffer; note that Y is a function of X
X
Y
pk
0
0
1
0
2
0
3
1
4
2
1
9
2
9
2
9
2
9
2
9
E[Y ] = 0 · p0 + 1 · p1 + 2 · p2 + 3 · p3 + 4 · p4
2
2
2
2
1
= 0· +0· +0· +1· +2·
9
9
9
9
9
= 5/9
2