**CHAPTER 6 6.2. STATIONARY DISTRIBUTION**

**6.2 Stationary distribution**

In this section, we first develop the stationary differential difference equations for joint
distributions of server state, orbit length and elapsed service time (or the elapsed retrial
time). We define the partial PGFs on the basis of the server state *C*(*t*). The probability
that the orbit is empty and the system is in WV at time *t* is

*P*0(*t*) =*P{C*(*t*) = 0*, N*(*t*) = 0*}, t* *≥*0*.*

For*n* *≥*1,

*P*_{n}(*t, x*)*dx*=*P{C*(*t*) = 0*, N*(*t*) =*n, x < ξ*(*t*)*≤x*+*dx}, t≥*0*,*

which is the probability that at time *t* the system is idle in WV with *n* customers in the
orbit and the elapsed retrial time of the customer at the head of the orbit lies between*x*
and *x*+*dx*. Similarly, for the idle system in non-vacation period, we get

*H*_{0}(*t*) =*P{C*(*t*) = 1*, N*(*t*) = 0*}, t* *≥*0*,*

*H*_{n}(*t, x*)*dx*=*P{C*(*t*) = 1*, N*(*t*) = *n, x < ξ*(*t*)*≤x*+*dx}, t≥*0*, n≥*1*.*

Also,

*S*_{n}(*t, x*)*dx*=*P{C*(*t*) = 2*, N*(*t*) = *n, x < ξ*(*t*)*≤x*+*dx}, t≥*0*, n≥*0*, x >*0*,*
which is the probability that at time *t* the server is busy in WV with *n* customers in the
orbit and the elapsed service time of the customer under service lies between*x*and*x*+*dx*.

*CHAPTER 6* *6.2. STATIONARY DISTRIBUTION*

For the busy system in non-vacation period, we define similarly,

*W*_{n}(*t, x*)*dx* =*P{C*(*t*) = 3*, N*(*t*) = *n, x < ξ*(*t*)*≤x*+*dx}, t≥*0*, n≥*0*, x >*0*.*

By considering transitions of the process between *t* and *t*+ ∆*t* and letting ∆*t* *→* 0, we
derive the following system of Kolmogorov’s forward equations

· *d*
*dt* +*λ*

¸

*P*0(*t*) =
Z _{∞}

0

*µ**v*(*x*)*S*0(*t, x*)*dx*+
Z _{∞}

0

*µ**b*(*x*)*W*0(*t, x*)*dx,*
(6.4)

·*∂*

*∂t*+ *∂*

*∂x* +*λ*+*θ*+*α*(*x*)

¸

*P**n*(*t, x*) = 0*,* *n* *≥*1*,* (6.5)

*H*0(*t*) = 0*.* (6.6)

·*∂*

*∂t*+ *∂*

*∂x* +*λ*+*α*(*x*)

¸

*H**n*(*t, x*) = 0*,* *n* *≥*1*,* (6.7)

·*∂*

*∂t* + *∂*

*∂x* +*λ*+*θ*+*µ**v*(*x*)

¸

*S*0(*t, x*) = *λ*(1*−a*)*S*0(*t, x*)*,* (6.8)

·*∂*

*∂t*+ *∂*

*∂x* +*λ*+*θ*+*µ**v*(*x*)

¸

*S**n*(*t, x*) = *λaS**n−*1(*t, x*) +*λ*(1*−a*)*S**n*(*t, x*)*,* *n≥*1*,* (6.9)

·*∂*

*∂t*+ *∂*

*∂x* +*λ*+*µ**b*(*x*)

¸

*W*0(*t, x*) = *θS*0(*t, x*)*,* (6.10)

· *∂*

*∂t*+ *∂*

*∂x* +*λ*+*µ**b*(*x*)

¸

*W**n*(*t, x*) = *λW**n−*1(*t, x*) +*θS**n*(*t, x*)*,* *n* *≥*1*.* (6.11)
These equations are to be solved under the boundary conditions at*x*= 0 which includes
*P**n*(*t,*0)*, H**n*(*t,*0)*, S*0(*t,*0)*, S**n*(*t,*0)*, W*0(*t,*0) and*W**n*(*t,*0). We elaborate the expressions
for these boundary conditions as follows:

1. *P*_{n}(*t,*0) is the probability of the event that at time*t*, the server is idle in WV, with
exactly *n* customers in the orbit and the customer at the head of the orbit has
just started its retrial. This event occurs when, at a service completion epoch, the
customer leaves the system idle in WV with*n* customers in the orbit. Thus, we get

*P*_{n}(*t,*0) =
Z _{∞}

0

*µ*_{v}(*x*)*S*_{n}(*t, x*)*dx, n* *≥*1*.*

2. *H*_{n}(*t,*0) is the probability of the event that at time *t*, the system is idle in non-
vacation, with *n* customers in orbit and the customer at the head of the orbit has

*CHAPTER 6* *6.2. STATIONARY DISTRIBUTION*

just started its retrial. Such an event occurs when, at the service completion epoch,
the customer leaves the system idle in non-vacation with *n* customers in the orbit.

Therefore,

*H*_{n}(*t,*0) =
Z _{∞}

0

*µ*_{b}(*x*)*W*_{n}(*t, x*)*dx, n≥*1*.*

3. *S*_{0}(*t,*0) is the probability of the event that the system is in WV with an empty
orbit at service completion epoch *t*. Such an event can occur if one of the two
following cases happen. First is that the system is in WV with an empty orbit while
one customer arrives and gets service immediately. The other case is that the only
customer in the orbit retries successfully, leaving behind an empty orbit. Hence,

*S*_{0}(*t,*0) = *λP*_{0}(*t*) +
Z _{∞}

0

*α*(*x*)*P*_{1}(*t, x*)*dx.*

4. *S*_{n}(*t,*0) is the probability of the event that the system is in WV and there are exactly
*n* customers in the orbit at the service completion epoch*t*. Such an event can occur
if either of the following two events happen. First is, when the system is in WV
with *n* customers in the orbit and the arriving customer gets service immediately
upon his arrival. The second event is when there are (*n*+ 1) customers in the orbit,
the system is in WV and the retrial of a customer to get service is successful. Thus,

*S*_{n}(*t,*0) = *λ*
Z _{∞}

0

*P*_{n}(*t, x*)*dx*+
Z _{∞}

0

*α*(*x*)*P*_{n+1}(*t, x*)*dx, n≥*1*.*

5. *W*_{0}(*t,*0) is the probability of the event that the system is in non-vacation with
an empty orbit at the service completion epoch *t*. This is equivalent to the event
that when the system is in non-vacation and the only customer in the orbit retries
successfully, leaving behind an empty orbit. We obtain

*W*0(*t,*0) =
Z _{∞}

0

*α*(*x*)*H*1(*t, x*)*dx.*

6. *W**n*(*t,*0) is the probability of the event that there are exactly *n* customers in the
orbit at the service completion epoch *t*, with the system in non-vacation. Such an
event occurs if either of the following two events happen. The first event is that

*CHAPTER 6* *6.2. STATIONARY DISTRIBUTION*

there are*n* customers in the orbit with the system in non-vacation and the arriving
customer gets service immediately. The other event is that there are*n*+1 customers
in the orbit and the retrial of a customer to get service is successful. So,

*W*_{n}(*t,*0) = *λ*
Z _{∞}

0

*H*_{n}(*t, x*)*dx*+
Z _{∞}

0

*α*(*x*)*H*_{n+1}(*t, x*)*dx,* *n≥*1*.*

Thus, under these boundary conditions we have to solve the Kolmogorov’s forward equa- tions (6.4) to (6.11). We also have the normalization condition for the system given by

*P*_{0}(*t*) +*H*_{0}(*t*) +
X*∞*

*n*=1

Z _{∞}

0

[*P*_{n}(*t, x*) +*H*_{n}(*t, x*)]*dx*+
X*∞*

*n*=0

Z _{∞}

0

[*S*_{n}(*t, x*) +*W*_{n}(*t, x*)]*dx*= 1*.*

We assume that the stability condition, *λa < µ*_{b}, holds and that the stationary be-
havior of the system can be analyzed by defining

*P*_{0} = lim

*t→∞**P*_{0}(*t*)*,* *H*_{0} = lim

*t→∞**H*_{0}(*t*)*,*
*P*_{n}(*x*) = lim

*t→∞**P*_{n}(*t, x*)*,* *H*_{n}(*x*) = lim

*t→∞**H*_{n}(*t, x*)*,*
*S*_{n}(*x*) = lim

*t→∞**S*_{n}(*t, x*)*,* *W*_{n}(*x*) = lim

*t→∞**W*_{n}(*t, x*)*.*

The Kolmogorov’s forward equations for *t* *→ ∞*become
*λP*_{0} =

Z _{∞}

0

*µ*_{v}(*x*)*S*_{0}(*x*)*dx*+
Z _{∞}

0

*µ*_{b}(*x*)*W*_{0}(*x*)*dx.* (6.12)

· *d*

*dx* +*λ*+*θ*+*α*(*x*)

¸

*P*_{n}(*x*) = 0*, n≥*1*,* (6.13)

*H*_{0} = 0*,* (6.14)

· *d*

*dx* +*λ*+*α*(*x*)

¸

*H*_{n}(*x*) = 0*, n≥*1*,* (6.15)

· *d*

*dx* +*λ*+*θ*+*µ*_{v}(*x*)

¸

*S*_{0}(*x*) = *λ*(1*−a*)*S*_{0}(*x*)*,* (6.16)

· *d*

*dx* +*λ*+*θ*+*µ*_{v}(*x*)

¸

*S*_{n}(*x*) = *λaS*_{n−1}(*x*) +*λ*(1*−a*)*S*_{n}(*x*)*, n≥*1*,* (6.17)

· *d*

*dx* +*λ*+*µ*_{b}(*x*)

¸

*W*_{0}(*x*) = *θS*_{0}(*x*)*,* (6.18)

· *d*

*dx* +*λ*+*µ*_{b}(*x*)

¸

*W*_{n}(*x*) = *λW*_{n−1}(*x*) +*θS*_{n}(*x*)*, n≥*1*,* (6.19)

*CHAPTER 6* *6.2. STATIONARY DISTRIBUTION*

with the boundary conditions
*P*_{n}(0) =

Z _{∞}

0

*µ*_{v}(*x*)*S*_{n}(*x*)*dx, n≥*1 (6.20)
*H*_{n}(0) =

Z _{∞}

0

*µ*_{b}(*x*)*W*_{n}(*x*)*dx, n* *≥*1 (6.21)

*S*0(0) = *λP*0+
Z _{∞}

0

*α*(*x*)*P*1(*x*)*dx,* (6.22)

*S**n*(0) = *λ*
Z _{∞}

0

*P**n*(*x*)*dx*+
Z _{∞}

0

*α*(*x*)*P**n*+1(*x*)*dx, n* *≥*1 (6.23)
*W*0(0) =

Z _{∞}

0

*α*(*x*)*H*1(*x*)*dx,* (6.24)

*W**n*(0) = *λ*
Z _{∞}

0

*H**n*(*x*)*dx*+
Z _{∞}

0

*α*(*x*)*H**n*+1(*x*)*dx, n≥*1*,* (6.25)
and the normalizing equation is

*P*_{0}+*H*_{0}+
X*∞*

*n*=1

Z _{∞}

0

[*P*_{n}(*x*) +*H*_{n}(*x*)]*dx*+
X*∞*

*n*=0

Z _{∞}

0

[*S*_{n}(*x*) +*W*_{n}(*x*)]*dx* = 1*.*

We use the method of PGFs to solve these equations. First, we define the following partial PGFs

*Q*_{P}(*z, x*) =
X*∞*

*n*=1

*z*^{n}*P*_{n}(*x*)*,* *Q*_{H}(*z, x*) = *H*_{0}+
X*∞*

*n*=1

*z*^{n}*H*_{n}(*x*)*,*
*Q**S*(*z, x*) =

X*∞*

*n*=0

*z*^{n}*S**n*(*x*)*,* *Q**W*(*z*) =
X*∞*

*n*=0

*z*^{n}*W**n*(*x*)*,* *|z| ≤*1*.*

and integrating with respect to *x*, we get
*Q*_{P}(*z*) =

Z _{∞}

0

*Q*_{P}(*z, x*)*dx,*
*Q*_{H}(*z*) =

Z _{∞}

0

*Q*_{H}(*z, x*)*dx,*
*Q*_{S}(*z*) =

Z _{∞}

0

*Q*_{S}(*z, x*)*dx,*
*Q*_{W}(*z*) =

Z _{∞}

*n*=0

*Q*_{W}(*z, x*)*dx.*

*CHAPTER 6* *6.2. STATIONARY DISTRIBUTION*

Theorem 6.2.1. *The joint stationary distribution of the server state and the queue length*
*has the partial PGFs given by*

*Q*_{P}(*z*) = *λP*_{0}*R*^{∗}(*λ*+*θ*)

·*v*^{∗}(*λa*(1*−z*) +*θ*)*−K*_{1}*v*^{∗}(*λa*+*θ*)
1*−K*2*v*^{∗}(*λa*(1*−z*) +*θ*)

¸

*,* (6.26)

*Q*_{H}(*z*) = *λP*_{0}*θR*^{∗}(*λ*)

·*g*_{z}^{∗}(*λ*(1*−z*))*{*1*−K*_{1}*K*_{2}*v*^{∗}(*λa*+*θ*)*}*

1*−K*2*v*^{∗}(*λa*(1*−z*) +*θ*) *−K*_{1}*g*^{∗}_{0}(*λ*)

¸
*,*(6.27)

*Q*_{S}(*z*) = *λP*_{0}*V*^{∗}(*λa*(1*−z*) +*θ*)

· 1*−K*_{1}*K*_{2}*v*^{∗}(*λa*+*θ*)
1*−K*2*v*^{∗}(*λa*(1*−z*) +*θ*)

¸

*,* (6.28)

*and* *Q*_{W}(*z*) = *λθP*_{0}*g*^{∗}_{z}(*λ*(1*−z*))

· 1*−K*_{1}*K*_{2}*v*^{∗}(*λa*+*θ*)
1*−K*2*v*^{∗}(*λa*(1*−z*) +*θ*)

¸

*,* (6.29)

*where*

*K*1 = 1

*v*^{∗}(*λa*+*θ*) +*θg*_{0}^{∗}(*λ*)*,* *K*2 =*λR*^{∗}(*λ*+*θ*) +*r*^{∗}(*λ*+*θ*)*,* *and*
*g*^{∗}_{z}(*λ*(1*−z*)) =

Z _{∞}

0

*e*^{−λ(1−z)x}*b*(*x*)

·Z _{x}

0

*e**−*[*θ−λ*(1*−a*)(1*−z*)]*t**V*(*t*)
*B*(*t*)*dt*

¸
*dx.*

*Proof.* Multiplying equation (6.13) with *z*^{n} and summing over *n* = 1*,*2*, . . . ,* we obtain

*∂*

*∂xQ*_{P}(*z, x*) + (*λ*+*θ*+*α*(*x*))*Q*_{P}(*z, x*) = 0*,*
which implies that

*Q*_{P}(*z, x*) = *Q*_{P}(*z,*0)*e*^{−(λ+θ)x}*R*(*x*)*.* (6.30)
Similarly, from equation (6.15) we have

*Q**H*(*z, x*) =*Q**H*(*z,*0)*e*^{−λx}*R*(*x*)*.* (6.31)
Equations (6.16) and (6.17) yield

*Q*_{S}(*z, x*) =*Q*_{S}(*z,*0)*e**−*[*λa*(1*−z*)+*θ*]*x**V*(*x*)*.* (6.32)
Equations (6.18) and (6.19) give rise to a non-homogeneous equation

*∂*

*∂xQ*_{W}(*z, x*) + [*λ*(1*−z*) +*µ*_{b}(*x*)]*Q*_{W}(*z, x*) =*θQ*_{S}(*z, x*)

*CHAPTER 6* *6.2. STATIONARY DISTRIBUTION*

with the integrating factor *e*^{λ(1−z)x}_{B(x)}^{1} *,* which after solving reduces to
*Q**W*(*z, x*) =*θQ**S*(*z,*0)*e*^{−λ(1−z)x}*B*(*x*)

Z _{x}

0

*e**−*[*θ−λ*(1*−a*)(1*−z*)]*t**V*(*t*)

*B*(*t*)*dt.* (6.33)
The initial conditions (6.20)–(6.25) give rise to the PGFs

*Q**P*(*z,*0) =
Z _{∞}

0

*µ**v*(*x*)*Q**S*(*z, x*)*dx−*
Z _{∞}

0

*µ**v*(*x*)*S*0(*x*)*dx,* (6.34)
*Q**H*(*z,*0) =

Z _{∞}

0

*µ**b*(*x*)*Q**W*(*z, x*)*dx−*
Z _{∞}

0

*µ**b*(*x*)*W*0(*x*)*dx,* (6.35)

*Q*_{S}(*z,*0) = *λP*_{0}+*λ*
Z _{∞}

0

*Q*_{P}(*z, x*)*dx*+
Z _{∞}

0

*α*(*x*)*Q*_{P}(*z, x*)*dx,* (6.36)
*Q*_{W}(*z,*0) = *θQ*_{S}(*z,*0) +*λ*

Z _{∞}

0

*Q*_{H}(*z, x*)*dx*+
Z _{∞}

0

*α*(*x*)*Q*_{H}(*z, x*)*dx.* (6.37)
From (6.16), we have

*S*_{0}(*x*) =*S*_{0}(0)*e*^{−(λa+θ)x}*V*(*x*)*.* (6.38)
And, from (6.18) and (6.38)

*W*_{0}(*x*) =*θS*_{0}(0)*e*^{−λx}*B*(*x*)
Z _{x}

0

*e**−*[*θ−λ*(1*−a*)]*t**V*(*t*)

*B*(*t*)*dt.* (6.39)

Inserting (6.38) and (6.39) in (6.12), we obtain
*S*_{0}(0) = *λP*_{0}

*v*^{∗}(*λa*+*θ*) +*θg*_{0}^{∗}(*λ*) =*λP*_{0}*K*_{1}*,* (6.40)
where

*K*_{1} = 1

*v*^{∗}(*λa*+*θ*) +*θg*_{0}^{∗}(*λ*)
and

*g*_{0}^{∗}(*λ*) =
Z _{∞}

0

*e*^{−λx}*b*(*x*)

·Z _{x}

0

*e**−*[*θ−λ*(1*−a*)]*t**V*(*t*)
*B*(*t*)*dt*

¸
*dx.*

Using (6.32) and (6.40) in (6.34), we get

*Q**P*(*z,*0) = *Q**S*(*z,*0)*v*^{∗}(*λa*(1*−z*) +*θ*)*−λP*0*K*1*v*^{∗}(*λa*+*θ*)*.* (6.41)
Again, using the above equation in (6.36), we get

*Q**S*(*z,*0) = *λP*0[1*−K*1*K*2*v*^{∗}(*λa*+*θ*)]

1*−K* *v*^{∗}(*λa*(1*−z*) +*θ*)*,* (6.42)

*CHAPTER 6* *6.2. STATIONARY DISTRIBUTION*

where

*K*_{2} =*λR*^{∗}(*λ*+*θ*) +*r*^{∗}(*λ*+*θ*)*.*

Therefore, from (6.32), we find that

*Q**S*(*z, x*) = *λP*0[1*−K*1*K*2*v*^{∗}(*λa*+*θ*)]*e**−*[*λa*(1*−z*)+*θ*]*x**V*(*x*)

1*−K*_{2}*v*^{∗}(*λa*(1*−z*) +*θ*) (6.43)
and, from (6.30), we find that

*Q*_{P}(*z, x*) =

"

*λP*_{0}*v*^{∗}(*λa*(1*−z*) +*θ*)*{*1*−K*_{1}*K*_{2}*v*^{∗}(*λa*+*θ*)*}*

1*−K*2*v*^{∗}(*λa*(1*−z*) +*θ*)

*−λP*_{0}*K*_{1}*v*^{∗}(*λa*+*θ*)

#

*e*^{−(λ+θ)x}*R*(*x*)

= *λP*0

·*v*^{∗}(*λa*(1*−z*) +*θ*)*−K*_{1}*v*^{∗}(*λa*+*θ*)
1*−K*_{2}*v*^{∗}(*λa*(1*−z*) +*θ*)

¸

*e*^{−(λ+θ)x}*R*(*x*)*.* (6.44)
In a similar fashion, the other generating functions can be derived as

*Q*_{H}(*z, x*) = *λθP*_{0}

·*g*^{∗}_{z}(*λ*(1*−z*))*{*1*−K*_{1}*K*_{2}*v*^{∗}(*λa*+*θ*)*}*

1*−K*2*v*^{∗}(*λa*(1*−z*) +*θ*) *−K*_{1}*g*_{0}^{∗}(*λ*)

¸

*e*^{−λx}*R*(*x*)*,*

(6.45)
*Q*_{W}(*z, x*) = *λθP*_{0}

· 1*−K*_{1}*K*_{2}*v*^{∗}(*λa*+*θ*)
1*−K*2*v*^{∗}(*λa*(1*−z*) +*θ*)

¸

*e*^{−λ(1−z)x}*B*(*x*)
Z _{x}

0

*e**−*[*θ−λ*(1*−a*)(1*−z*)]*t**V*(*t*)
*B*(*t*)*dt,*

(6.46) where

*g*_{z}^{∗}(*λ*(1*−z*)) =
Z _{∞}

0

*e*^{−λ(1−z)x}*b*(*x*)

·Z _{x}

0

*e**−*[*θ−λ*(1*−a*)(1*−z*)]*t**V*(*t*)
*B*(*t*)*dt*

¸
*dx.*

Integrating equations (6.43)-(6.46) with respect to *x*, we get the desired partial PGFs
(6.26)-(6.29).

All these partial PGFs are in terms of the unknown quantity*P*_{0}. The derivation of*P*_{0}
using the normalization equation is given in the next theorem.

Theorem 6.2.2. *The probability* *P*_{0} *is given by the following expression*
*P*_{0} = [1*−K*_{2}*v*^{∗}(*θ*)]

.h

*λR*^{∗}(*λ*+*θ*)*{v*^{∗}(*θ*)*−K*_{1}*v*^{∗}(*λa*+*θ*)*}*+*λθR*^{∗}(*λ*)*{g*^{∗}_{1}(0)*−K*_{1}*g*_{0}^{∗}(*λ*)*}*

+*λθR*^{∗}(*λ*)*K*_{1}*K*_{2}*{g*_{0}^{∗}(*λ*)*v*^{∗}(*θ*)*−g*_{1}^{∗}(0)*v*^{∗}(*λa*+*θ*)*}*+ [1*−K*_{2}*v*^{∗}(*θ*)]

+*λ*
n

*V*^{∗}(*θ*) +*θg*_{1}^{∗}(0)
o

*{*1*−K*_{1}*K*_{2}*v*^{∗}(*λa*+*θ*)*}*

i

*.* (6.47)