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**Unformatted text preview: **th Processes Examples Birth-Death Processes
For n = 0
P0 (t + h) = P0 (t )(1 − λ0 h) + P1 (t )µ1 h + o(h)
⇒ P0 (t ) = −λ0 P0 (t ) + µ1 P1 (t )
• If λ0 = 0, then E0 → E1 is impossible and E0 is an absorbing state.
• If λ0 = 0, then P0 (t ) = µ1 P1 (t ) ≥ 0 and hence P0 (t ) increases monotonically.
Note:
limt →∞ P0 (t ) = P0 (∞) = Probability of being absorbed.
18 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Steady-state distribution
P0 (t ) = −λ0 P0 (t ) + µ1 P1 (t )
Pn (t ) = −(λn + µn )Pn (t ) + λn−1 Pn−1 (t ) + µn+1 Pn+1 (t )
As t → ∞, Pn (t ) → Pn (limit ).
Hence, P0 (t ) → 0 and Pn (t ) → 0.
Therefore,
0 = −λ0 P0 + µ1 P1
λ0
⇒P1 =
P0
µ1
0 = −(λ1 + µ1 )P1 + λ0 P0 + µ2 P2
λ0 λ1
⇒P2 =
P0
µ1 µ2
λ0 λ1 λ2
⇒P3 =
P0
etc
µ1 µ2 µ 2 19 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Steady-state distribution P1 = λ0
P0 ;
µ1 P2 = λ0 λ1
P0 ;
µ 1 µ2 P3 = λ0 λ1 λ2
P0 ;
µ1 µ2 µ 2 P4 = . . . The dependence on the initial conditions has disappeared.
∞ After normalizing, i.e., Pn = 1:
n =1
n−1 1 P0 = ∞ n −1 1+
n=1 i =0 ;
λi
µi +1 λi
µ i +1
i =0
∞ n −1 Pn =
1+ n=1 i =0 , n≥1
λi
µi +1 20 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Steady-state distribution n−1 1 P0 = ∞ n −1 1+
n=1 i =0 ; i =0
∞ n −1 Pn = λi
µi +1 λi
µ i +1 1+
n=1 i =0 , n≥1
λi
µi +1 Ergodicity condition
Pn > 0, for all n ≥ 0, i.e.,:
∞ n−1
n=1 i =0 λi
<∞
µ i +1 21 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Example. A single server system / Birth-death processes
J. Virtamo
38.3143 Queueing Theory
Example 3. A single server system
l
0 - constant arrival rate λ (Poisson arrivals)
• constant arrival rate λ (Poisson
- stopping rate of the service µ (exponential distribution
arrivals) 1
m The states of the system • stopping rate of service µ 0
server free
(exponential distribution) 1
server busy
• states of the system: 0 (server 1 ì
í
î ì
í
î 0 free), 1 (server busy) ~ Exp(m) ~ Exp(l) d
dt π0(t) = − λπ0(t) + µπ1(t) d
dt π1(t) = P0 (t ) = −λP0 (t ) + µP1=t ) −λ
Q( λπPt)(− µπ1λtP (t ) − µP (t )
0( t ) = ( )
0
1
1
BY adding both sides of the equations
d
dt (π0 (t)
d + π1(t)) = 0 ⇒ π (t) + (λ + µ)π (t) = µ µ π0 (t) + π1 (t) = constant = 1
⇒ d λ
−µ ⇒ (e(λ+µ)t π (t)) = µe(λ+µ)t π1(t) = 1 − π0 (t
22 / 47 Birth Processes Birth-Death Processes Relationship to Markov Chains J. Virtamo Linear Birth-Death Processes 38.3143 Queueing Theory / Birth-death processes Example. 3. A single server system
A single server system
Example Examples 7 l
0 - constant arrival rate λ (Poisson arrivals)
- stopping rate of the service µ (exponential distribution) 1
m The states of theP (t )
system 0 0
server free P1 (
1
server b...

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