Chapter 3. Analysis of an Un-Reliable Server M [x] /G/1 Queueing System
3.5 Reliability Indices
In this section, we develop two main reliability indices of the presented model, namely, the system availability and failure frequency under the steady state conditions. Let us define Av(t) as the system availability at time t, that is, the probability that the server is either working for a customer or remaining idle in the system. The steady state availability of the server is given by Av = limt→∞Av(t).
Corollary 3.1:
The steady-state availability of the server is given by
Av = ρ + 1− ρH
1 + λE[V ](1−(¯pγ0)J)
γ0
[
( ¯pγ0)J−1(1−¯pγ0)+p(1−(¯pγ0)J−1)
].
Proof:
We first consider the following equation Av = P0+
∫ ∞
0
P (x, 1)dx = P0+ lim
z→1P (z),
then using equations (3.31) and (3.37), we can get Corollary 3.1.
Corollary 3.2:
The steady-state failure frequency of the server is given by Mf = αρ.
Proof:
Following the argument by Li et al. [33], we obtain Mf = α
∫ ∞
0
P (x, 1)dx,
then using (3.1), we can get Corollary 3.2.
3.6 Some Special Cases of the Proposed Model
In this section, we demonstrate some existing results in the literature which are special cases of our system.
Case 3.1: Special one of our model is the ordinary M/G/1 queueing system with at most
which agrees with Ke [26].
Case 3.2: Letting p = 0, α = 0 and J = 1, our model can be simplified to the ordinary M/G/1 queueing system with single vacation. Φ(z) can be rewritten as
((1− ρ)(1 − z)S∗(a(z)) which confirms the result in Section 6 of Choudhury’s system [5].
Case 3.3: Letting p = 0 and if we do not consider the randomized vacation policy, Φ(z) can be expressed as
which is in accordance with Choudhury and Deka [6] for no optional service.
Case 3.4: Substituting p = 0, α = 0 and J =∞ into (3.38), Φ(z) can be expressed as
which is in accordance with Lee et al. [31] (multiple vacations) for no N-policy.
Case 3.5: Suppose that we have p = 0, α = 0 and J = 1, then if we put Pr(X=1)=1, our model can be reduced to the ordinary M/G/1 queueing system with single vacation. It follows from (3.40) that the expected waiting time in the system is given by
Ws = λE[V2] 2(
λE[V ] + γ0) + λE[S2] 2(1− ρ),
which is in accordance with Takagi’s system [38].
Case 3.6: Letting p = 0 and α = 0, our model can be recovered to the M[x]/G/1 queueing system with at most J vacations. Using (3.40) we have the expected waiting time in the system as:
which is in accordance with Ke and Chu [28].
Case 3.7: If we consider the ordinary M/G/1 system with server breakdown without vacations (i.e., Pr(X=1)=1 and Pr(V=0)=1), (3.44) can be reexpressed as
Φ+(z) = (1− ρH)(1− z)S∗(A(z)) S∗(A(z))− z ,
which is consistent with Choudhury and Deka [6] for no optional service.
Case 3.8: Suppose that we have p = 0 and α = 0; then if we put J = 1, our system can be reduced to the ordinary M[x]/G/1 single vacation policy queue. In this case, φ(z) can be rewritten as
V∗(a(z)) + γ0[X(z)− 1],
which is in accordance with Choudhury’s system [5].
For convenient to compare our model with the existing literature, we summarize the results listed above in Table 3.1.
3.7 Optimal Randomized Operating Policy
In this section, We develop the long-run expected cost function per unit time for the M[x]/(G1,G2)/1/VAC(J) queueing system, in which p and J are the decision variables. Our objective is to determine the suitable values of the control variables p and J , say p∗ and J∗, so as to minimize the cost function. Let us define the following cost elements:
Ch≡ holding cost per unit time per customer present in the system;
Cs≡ set-up cost per busy cycle.
By using the renewal reward theory, we know that the long-run expected cost per unit time is given by
F (p, J ) = ChLs+ Cs E[C]
= A1+ A2(1− (¯pγ0)J) + A3(1− ¯pγ0) B1(1− (¯pγ0)J) + γ0(
p(1− (¯pγ0)J−1) + (¯pγ0)J−1B2), (3.54)
where
For analysis, J may be treated as a continuous variable greater than zero. Noting that if J∗ is not an integer, the best positive integer value of J is one of the integers surrounding J∗. Differentiating F (p, J ) (in (3.54)) with respect to p and J respectively, it gives
∂F (p, J )
with
D11 = A2γ0JJ ¯pJ−1+ A3γ0,
D12 = B1(1− (¯pγ0)J) + γ0 (
p(1− (¯pγ0)J−1) + (¯pγ0)J−1(1− ¯pγ0) )
,
D13 = A2(1− (¯pγ0)J) + A3(1− ¯pγ0),
D14 =−B1γ0JJ ¯pJ−1+ γ0
(
(1− (¯pγ0)J−1) + p(J− 1)γ0J−1p¯J−2 )
, and
D15= γ0 (
− γ0J−1(J − 1)¯pJ−2(1− ¯pγ0) + (¯pγ0)J−1γ0 ))
.
For any given J , we know that:
1a. If D1 > 0 in p ∈ [0, 1], by using (3.55) yields ∂F (p,J )∂p > 0 which means F (p, J ) is an increasing function of p in p∈ [0, 1].
2a. If D1 < 0 in p ∈ [0, 1], by using (3.55) yields ∂F (p,J )∂p < 0 which implies F (p, J ) is a decreasing function of p in p∈ [0, 1].
3a. Noting that ∂F (p,J )∂p = 0 iff D1 = 0 (in this case, p∗ is a solver of D1 = 0). This case is a rare event (see the structure of D1). That is, the occurrence of D1 is very small.
For any given p, we also know that:
1b. If D2 > 0, by using (3.56) yields ∂F (p,J )∂J > 0 which means F (p, J ) is an increasing function of J .
2b. If D2 < 0, by using (3.56) yields ∂F (p,J )∂J < 0 which implies F (p, J ) is a decreasing function of J .
3b. Noting that ∂F (p,J )∂J = 0 iff D2 = 0 (in this case, J∗ is a solver of D2 = 0). Noting that the occurrence of the case D2 = 0 is very small.
In order to find the joint optimal values of p and J , say p∗ and J∗, we should solve the following equations:
∂F (p, J )
∂p = 0 and ∂F (p, J )
∂J = 0. (3.59)
The solutions (p, J ) = (p∗, J∗) attain a minimum if it satisfies the following
∂2F (p, J )
∂p2 > 0, (3.60)
∂2F (p, J )
∂J2 > 0, (3.61)
and the determinant of the Hessian matrix is positive definite, that is, det(H) = ∂2F (p, J )
∂p2 · ∂2F (p, J )
∂J2 −(∂2F (p, J )
∂p∂J )2
> 0. (3.62)
Noting that (3.59) is just necessary conditions for F (p, J ) to attain it’s minimum.
Although we cannot analytically prove that F (p, J ) is a convex function of (p, J ) indeed, one heuristic approach is provided to search the joint optimum values of p and J . By the inferences listed above we know that for a given p, the optimal values J∗ of J is J∗ = 1, arbitrary positive integer or J∗ = ∞ (say M, where M is a sufficiently large number in practice). A heuristic decision is summarized in the following that makes it possible to determine the joint suitable values (p∗, J∗) as follows:
• The criterion for searching the joint suitable thresholds p∗ and J∗: Case 1 If D1 > 0 and
i. D2 > 0 then (p∗, J∗) = (0, 1), ii. D2 < 0 then (p∗, J∗) = (0,∞),
iii. D2 = 0 then (p∗, J∗) = (0, J∗ is a solver of D2 = 0).
Case 2 If D1 < 0 and
i. D2 > 0 then (p∗, J∗) = (1, 1), ii. D2 < 0 then (p∗, J∗) = (1,∞),
iii. D2 = 0 then (p∗, J∗) = (1, J∗ is a solver of D2 = 0).
Case 3 If D1 = 0 and
i. D2 > 0 then (p∗, J∗)=(p∗ is a solver of D1 = 0 , 1), ii. D2 < 0 then (p∗, J∗)=(p∗ is a solver of D1 = 0, ∞),
iii. D2 = 0 then (p∗, J∗)=(p∗ is a solver of D1 = 0, J∗ is a solver of D2 = 0).
3.8 Numerical Illustration
This section we perform two extensive numerical examples to illustrate the joint suitable randomized behavior based on Section 3.7.
Firstly, we consider the case of D1 > 0 and D2 < 0 with the setting system’s parameters:
1. Customers arrive in batches occurring according to a compound Poisson process with mean arrival rate λ = 0.6;
2. Geometric batch size with mean E[X] = 2;
3. The 4-stage Erlang service time with mean service time per batch E[S] = 1;
4. The vacation time follows the exponential distribution with E[V ] = 1;
5. The Poisson breakdown rate α = 0.05;
6. The repair time follows the 2-stage Erlang distribution with E[R] = 0.2;
7. The holding cost Ch = 10 and the set-up cost Cs = 10000.
The expected cost F (p, J ) for this case is shown in Figure 3.1. Noting that the minimum cost per unit time of $1549.7 is achieved at p∗ = 0 and J∗ is 20 or more.
The second numerical example is the case of D1 > 0 and D2 > 0 with the following system’s parameters:
1. Customers arrive in batches to occur according to a compound Poisson process with mean arrival rate λ = 0.6;
2. Geometric batch size with mean E[X] = 3;
3. The 4-stage Erlang service time with mean service time per batch E[S] = 1;
4. The vacation time follows the exponential distribution with E[V ] = 0.5;
5. The Poisson breakdown rate α = 0.05;
6. The repair time follows the 2-stage Erlang distribution with E[R] = 0.2;
7. The holding cost Ch = 10 and the set-up cost Cs = 10000.
It is seen from Figure 3.2, for the D1 > 0 and D2 > 0 case, that a minimum cost per unit time of $409.3831 is achieved at p∗ = 0 and J∗ = 1.
The numerical results reveal that (i) the expected cost increases as p increases; and (ii) J decreases as p increases. On the other hand, the numerical results are also in accordance with the conclusion in Section 3.7. (i.e., the joint optimal values is (0, 1)-single vacation, or (0, M )-multiple vacation vacation, where M is a sufficiently large number for practice use). These special policies can be also referred to special cases 3.2 and 3.4. This implies the optimal vacation policy is exactly as single vacation or multiple vacation policy.
Table 3.1. Summary of some special cases of our model [M[x]/(G1, G2)/1/VAC(J)]
Our model Under what
condition Special case Reduce to what kind of model
(3.37) p=0andα =0. Ke [26, p.1326, Equation (25)]
Ordinary M[x]/G/1 queueing system with at most J vacations.
=0 queueing system with a single vacation.
queueing system with a single vacation. at most J vacations.
=0 and a delayed repair.
(3.44) queueing system with a single vacation. and a delayed repair.
0
20 40
60 80
100
0 0.5
1 1500 1600 1700 1800 1900 2000
p J
F(p,J)
F(p*, J*)=1549.7 where p* is 0 and J* is 20 or more
Figure 3.1. The expected cost for the different values p and J (D1 > 0, D2 < 0).
0
20 40
60 80
100
0 0.5
1 400 410 420 430 440 450 460
p J
F(p,J)
F(p*, J*)=409.3831 where p* is 0 and J* is 1
Figure 3.2. The expected cost for the different values p and J (D1 > 0, D2 > 0).
Chapter 4
Analysis of an Un-Reliable Server M
[x]/G/1 System with Randomized Vacation Policy and a Delayed
Repair
In Chapter 4, we extend the queueing models discussed in Chapters 2 and 3 to the un-reliable M[x]/G/1 queueing system, in which the un-reliable server operates a randomized vacation policy with at most J vacations. The server is subject to breakdowns at any time with a Poisson breakdown rate α when he is working. As soon as the server fails it is sent for repair during which the server stops providing to arriving customer and need to wait for repair to begin service. We define the waiting time of repair as delayed time.
This chapter is organized as follows: Basic assumptions and notations of the queueing system under study are given in Section 4.1. In Section 4.2, we develop the steady-state differential-difference equations for the M[x]/(G1,G2)/1/VAC(J)/(Delayed Repair) queueing system by treating the elapsed service time, the elapsed repair time, the elapsed waiting time for repair and the elapsed vacation time as supplementary variables. Then we solve these system equations and derive the probability generating functions (p.g.f’s) of various server’s states at a random epoch. In Section 4.3, using the p.g.f’s obtained in Section 4.2, we de-rive the expected number of customers in the system and the expected waiting time for the M[x]/(G1,G2)/1/VAC(J)/(Delayed Repair) system. Section 4.4 we develop some important system characteristics such as the queue size distribution at a departure epoch, the system size distribution at busy period initiation epoch and the busy period and idle period distribu-tion, etc. We also derive the Laplace Stieltjes Transform (LST) for the busy period and idle period, and further obtain the expected length of busy period, idle period and busy cycle. In Section 4.5, we develop two important reliability indices of the presented model, namely, the steady state system availability and steady state failure frequency. In Section 4.6, we demon-strate some existing literatures are special cases of our model. In Section 4.7, we develop the long-run expected cost function per unit time for the M[x]/(G1,G2)/1/VAC(J)/(Delayed Repair) queueing system and determine the suitable values of the control variables p and J ,
say p∗ and J∗, so as to minimize the cost function. Finally, in Section 4.8, some numerical experiments are performed to demonstrate the optimization behavior.
4.1 Assumptions and Notations
In this chapter, we consider the M[x]/(G1,G2)/1/VAC(J)/(Delayed Repair) queueing system with an un-reliable server and a delayed repair, in which the server operates a random-ized vacation policy and takes at most J vacations when he serves all customers exhaustively in the system. The detailed description of the model is given as follows: Customers arrive in batches according to a compound Poisson process with arrival rate λ. Let Xk denote the number of customers belonging to the kth arrival batch, where Xk, k=1,2,3,..., are with a common distribution
P r(Xk = n) = χn, n = 1, 2, 3, ...
The service time provided by a single server is an independent and identically dis-tributed random variable (S) with distribution function S(t) and Laplace-Stieltjes transform (LST) S∗(θ). Arriving customers who join the system form a single waiting line based on the order of their arrivals, that is, they are queued according to the first-come, first-served (FCFS) discipline. The server can serve only one customer at a time, and that the service is independent of the arrival of the customers. If the server is busy or on vacation, arrivals in the queue must wait until the server is available. When the system becomes empty, the server leaves for a vacation with random length V . Upon returning from the vacation, the server operates a randomized vacation policy with at most J vacations. If at least one customer is found waiting in the queue upon returning from the vacation, the server is immediately activated for service. Alternatively, if no customers are found in the queue at the end of a vacation, the server remains idle in the system with probability p and leaves for another vacation with probability ¯p (= 1− p). This pattern continues until the number of vacations taken reaches J. If the system is still empty by the end of the Jth vacation, the server remains idle in the system. If there is at least one customer arrives at server idle state, the server immediately starts providing his services for the arrivals. The vacation time V has distri-bution function V (t) and Laplace-Stieltjes transform (LST) V∗(θ). The server is subject to breakdowns at any time with Poisson breakdown rate α when he is working. As soon as the
server fails it is sent for repair during which the server stops providing to arriving customer and waits for repair to begin service. We define the waiting time of repair as delayed time, where the delayed time is an independent and identically distributed random variable D with a general distribution function D(t) and Laplace-Stieltjes transform (LST) D∗(θ). The repair time of the broken-down server is an independent and identically distributed random variable R with a general distribution function R(t) and Laplace-Stieltjes transform (LST) R∗(θ). A customer who arrives and finds the server busy or broken down must wait in the queue until a server is available. Although no service occurs during the repair period of a broken server, customers continue to arrive according to a compound Poisson process. In case the server breaks down when serving customers, the server is sent for repair and the customer who has just being served should wait for server back to complete his remaining service. Immediately after the server is fixed, he starts to serve customers until the system is empty and the service time is cummulative. Furthermore, various stochastic processes involved in the system are independent of each other.
Moreover, let us define G as the generalized service time random variable representing the completion of a customer service, which consists of both the service time of a customer and the repair time of a server, including a repair possibile delay. The LST of G can be expressed as follows:
G∗(θ) =
∫ ∞
0
∑∞ n=0
e−αt(αt)n
n! e−θt[D∗(θ)R∗(θ)]ndS(t),
= S∗(
θ + α(1− D∗(θ)R∗(θ)))
. (4.1)
From (4.1), we obtain the first moment of G given by E[G] =− d
dθ[G∗(θ)]|θ=0 = E[S](
1 + α(E[D] + E[R]))
, (4.2)
where E[S] = −dθd[S∗(θ)]|θ=0 is the mean service time, E[D] = −dθd[D∗(θ)]|θ=0 is the mean delayed time and E[R] =−dθd[R∗(θ)]|θ=0 is the mean repair time.
In this chapter, the following notations are used.
1. λ ≡ Poisson arrival rate.
2. Xk ≡ the number of customers belonging to the kth arrival batch.
3. α ≡ Poisson breakdown rate.
4. S, S(t), S∗(θ) ≡ service time random variable, service time distribution function and LST of S(t), respectively.
5. D, D(t), D∗(θ)≡ delayed time random variable, delayed time distribution function and LST of D(t), respectively.
6. R, R(t), R∗(θ) ≡ repair time random variable, repair time distribution function and LST of R(t), respectively.
7. V, V (t), V∗(θ) ≡ vacation time random variable, vacation time distribution function and LST of V (t), respectively.
8. µ(x)dx ≡ the first order differential (hazard rate) functions of S.
9. ω(x)dx ≡ the first order differential (hazard rate) functions of V . 10. N (t) ≡ the number of customers in the system.
11. S−(t) ≡ the elapsed service time.
12. D−(t) ≡ the elapsed waiting time for repair.
13. R−(t)≡ the elapsed repair time.
14. V−(t) ≡ the elapsed time of the jth vacation.
15. P0(t)≡ the probability that the server is idle but available at time t.
16. Pn(x, t)dx≡ the probability that there are n customers in the system when the server is busy at time t.
17. Wn(x, t)dx≡ the probability that there are n customers in the system when the server is waiting for repair at time t given that the elapsed service time is x.
18. Qn(x, t)dx ≡ the probability that there are n customers in the system when the server is under repair at time t given that the elapsed service time is x.
19. Ωj,n(x, t)dx≡ the probability that there are n customers in the system when the server is on the jth vacation at time t.
20. P0 ≡ the steady state probability that the server is idle but available at.
21. Pn(x) ≡ the steady state probability that there are n customers in the system when the server is busy.
22. Wn(x) ≡ the steady state probability that there are n customers in the system when the server is waiting for repair given that the elapsed service time is x.
23. Qn(x) ≡ the steady state probability that there are n customers in the system when the server is under repair given that the elapsed service time is x.
24. Ωj,n(x) ≡ the steady state probability that there are n customers in the system when the server is on the jth vacation.
25. γ0 ≡ the probability that there are no batches arrive during a vacation time.
4.2 Development of the Equations and Solutions
In this section, we first develop the steady-state differential-difference equations for the variant vacation system by treating the elapsed service time, the elapsed repair time, the elapsed delayed time and the elapsed vacation time as supplementary variables. Then we solve these system equations and derive the probability generating functions of various server states at a random epoch.
We define the state of the system at time t as follows:
N (t)≡ the number of customers in the system, S−(t) ≡ the elapsed service time,
R−(t) ≡ the elapsed repair time,
D−(t) ≡ the elapsed waiting time for repair, and
Vj−(t) ≡ the elapsed time of the jth vacation.
We introduce the following random variable for further development of the variant vacation queueing model:
∆(t)=
0, if the server is idle in the system at time t, 1, if the server is busy at time t,
2, if the server is under repair at time t, 3, if the server is waiting for repair at time t, 4, if the server is on the 1th vacation at time t,
Furthermore, let us define the following probabilities:
P0(t) = Pr{N(t) = 0, δ(t) = 0},
ξ(y)dy = 1dD(y)−D(y) and ω(x)dx = 1dV (x)−V (x) are the first order differential functions (hazard rate) of S, R, D and V , respectively.
In steady-state, we can set P0 = limt→∞P0(t); Pn(x) = limt→∞Pn(x, t); Qn(x, y) = limt→∞Qn(x, y, t); Wn(x, y) = limt→∞Wn(x, y, t) and Ωj,n(x) = limt→∞Ωj,n(x, t). According
to Cox [10], the Kolmogorov forward equations that govern the system under steady-state conditions can be written as follows:
λP0 =
The above equations are to be solved by means of the following boundary conditions at x = 0
Pn(0) =
with normalization condition
where a(z) = λ(1− X(z)). Similarly, proceeding in the usual manner with (4.5)-(4.9), we have
where x > 0.
Solving the partial differential equations (4.15)-(4.18), we obtain
P (x; z) = P (0; z)[1− S(x)]e−A(z)x, (4.20)
Q(x, y; z) = Q(x, 0; z)[1− R(y)]e−a(z)y, (4.21)
W (x, y; z) = W (x, 0; z)[1− D(y)]e−a(z)y, (4.22)
and
Ωj(x; z) = Ωj(0; z)[1− V (x)]e−a(z)x, j = 1, 2, ...J, (4.23) where A(z) = a(z) + α(
1− R∗(a(z))D∗(a(z))) . Solving the differential equation (4.7) yields
Ωj,0(x) = Ωj,0(0)[1− V (x)]e−λx, j = 1, 2, ...J. (4.24) Now, (4.24) is multiplied by ω(x) on both sides for j = J and integrating with over x from 0 to ∞, we then have
∫ ∞
0
ΩJ,0(x)ω(x)dx = ΩJ,0(0)γ0, (4.25)
where γ0 = V∗(λ). Using (4.25) in (4.11), we can recursively obtain Ωj,0(0) = ΩJ,0(0)
(¯pγ0)J−j, j = 1, 2, ..., J − 1. (4.26)
Substituting (4.26) into (4.3) and after some algebraic manipulation, we have ΩJ,0(0) = λP0
γ0
[1 + ( ¯pγp(1−(¯pγ0)J−1)
0)J−1(1−¯pγ0)
]. (4.27)
From (4.26) and (4.27) we finally obtain Ωj(0; z) = Ωj,0(0) = λP0
(¯pγ0)J−jγ0[
1 + ( ¯pγp(1−(¯pγ0)J−1)
0)J−1(1−¯pγ0))
], j = 1, 2, ..., J. (4.28)
Integrating (4.24) with respect to x from 0 to∞ and using the well-known renewal theory
∫ ∞
0
e−λx(1− V (x))dx = 1− γ0
λ ,
we obtain
From (4.28) and (4.29), it finally yields Ωj,0= P0(1− γ0)
(¯pγ0)J−jγ0[
1 + ( ¯pγp(1−(¯pγ0)J−1)
0)J−1(1−¯pγ0)
], j = 1, 2, ..., J. (4.30)
Note that Ωj,0 represents the steady-state probability that there are no customers in the system when the server is on the jth vacation. Let us define Ω0 the probability that no customers appear in the system when the server is on vacation. Then we have
Ω0 = Inserting (4.20), (4.23) and (4.28) into (4.19) we get on simplification
P (0; z) = λP0(1− (¯pγ0)J)V∗(a(z))
Solving P (0; z) from (4.32) and using (4.28), yields
P (0; z) =
It follows from (4.20) and (4.33) that
P (x; z) =
Inserting (4.21) and (4.22) with the boundary conditions (4.12) and (4.13) respectively, W (x, 0; z) and Q(x, 0; z) can be expressed as
W (x, 0; z) = αP (x; z), (4.36)
and
Q(x, 0; z) = W (x, 0; z)D∗(a(z)). (4.37)
Utilizing (4.20), (4.36) in (4.22), we get
W (x, y; z) = αP (0; z)[1− S(x)]e−A(z)x[1− D(y)]e−a(z)y. (4.38) Similarly utilizing (4.20), (4.36) and (4.37) in (4.21), we have
Q(x, y; z) = αD∗(a(z))P (0; z)[1− S(x)]e−A(z)x[1− R(y)]e−a(z)y. (4.39) Inserting (4.33) into (4.38) and (4.39), we obtain
W (x, y; z) =
Calculating the double integral ∫∞
0
Q(z) = D∗(a(z))×
Using (4.23) and (4.28) we have Ωj(z) = P0(V∗(a(z))− 1)
[X(z)− 1](¯pγ0)J−jγ0
[1 + ( ¯pγp(1−(¯pγ0)J−1)
0)J−1(1−¯pγ0)
], j = 1, 2, 3, ...J. (4.44)
The unknown constant P0 can be determined by using the normalization condition (4.14), which is equivalent to P0+ P (1) + Q(1) + W (1) +∑J
Noting that (4.45) represents the steady state probability that the server is idle but available in the system. Also from (4.45), we have ρH < 1 which is the necessary and sufficient condition under which steady state solution exists.
Let Φ(z) = P0+P (z)+Q(z)+W (z)+∑J
j=1Ωj(z) be the probability generating function of the system size distribution at stationary point of time, we then have
Φ(z) = (1− ρH)S∗(a(z))(z− 1)
4.3 The Expected Number of Customers in the System and the Expected Waiting Time
In (4.46), we evaluate dzdΦ(z)|z=1 by using L’hopital rule which leads to the expected number of customers, Ls, in the system given by
Ls= ρH +
λE[X(X− 1)]E[G] +(
λE[X](
1 + α(E[D] + E[R])))2 E[S2] 2(1− ρH)
+α(λE[X])2(
(E[D2] + E[R2])E[S] + 2E[D]E[R]) 2(1− ρH)
+ λ2E[X](1− γ0J)E[V2] 2
(
λE[V ](1− (¯pγ0)J) + γ0[
(¯pγ0)J−1(1− ¯pγ0) + p(1− (¯pγ0)J−1)]). (4.47)
By using Little’s formula, we obtain the expected waiting time in the queue, Wq, given by
Wq = α(E[D] + E[R]) µ
+ E[X(X− 1)]E[G]
E[X](1− ρH)
+ λ
( E[X](
1 + α(E[D] + E[R])))2 E[S2] 2E[X](1− ρH)
+ αλE[X](
(E[D2] + E[R2])E[S] + 2E[D]E[R]) 2(1− ρH)
+ λ(1− γ0J)E[V2]
2 (
λE[V ](1− (¯pγ0)J) + γ0[
(¯pγ0)J−1(1− ¯pγ0) + p(1− (¯pγ0)J−1)]). (4.48)
4.4 Other System Characteristics
In this section, we first develop the queue size distribution at a departure epoch in sub-section 4.4.1, then the system size distribution at busy period initiation epoch is developed in subsection 4.4.2. Finally we derive the completion period and idle period distributions in subsection 4.4.3.
4.4.1 Queue Size Distribution at a Departure Epoch
We derive the probability generating function of the steady state distribution of the number of customers in the queue at a departure epoch for the queueing system under study. Following the arguments by Wolff [49], we state that a departing customer will see l customers in the queue just after a departure if and only if there are (l + 1) customers in the queue just before the departure. Thus we can write the following
Φ+l = K0
∫ ∞
0
µ(x)Pl+1(x)dx, l = 0, 1, ... (4.49)
where Φ+l = Pr(A departing customer will see l customers in the queue), and K0 is the nor-malizing constant.
Using the normalization condition Φ+(1) = 1, it gives
K0 = 1− ρ which leads to the probability generating function of the departure point queue size distri-bution as From (4.52), we observe that Φ+(z) can be decomposed into two independent terms:
Φ+(z) = 1− X(z)
E[X](1− z)× Φ(z). (4.53)
It should be noted that the departure point queue size distribution given by (4.53) can be decomposed into two independent random variables: One (the first term) is the number of
It should be noted that the departure point queue size distribution given by (4.53) can be decomposed into two independent random variables: One (the first term) is the number of