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 2.1: Special one of our model is the ordinary M^[x]/G/1 queueing system with at most J vacations. That is, if we let p = 0 in our model, then R₀ can be reduced to

1− ρ

λE[V ]

α^J₀ · ¹₁^−α_−α^J0₀ + 1 ,

which agrees with Ke [26].

Case 2.2: Letting p = 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 2.3: Letting p = 0, J =∞ and Pr(X=1)=1, our model becomes the ordinary M/G/1 queueing system with miltiple vacations. Φ(z) can be rewritten as

((1− ρ)(1 − z)S^∗(a(z)) and the result is in accordance with Takagi [38].

Case 2.4: Substituting p = 0 and J =∞ into (2.28), Φ(z) can be expressed as ((1− ρ)(1 − z)S^∗(a(z))

S^∗(a(z))− z

)( 1− V^∗(a(z)) (1− X(z))(

λE[V ]) )

,

which is in accordance with Lee et al. [31] (multiple vacations) for no N-policy.

Case 2.5: Suppose that we have p = 0, J = 1, then if we put P r(X = 1) = 1, our model can be reduced to the ordinary M/G/1 queueing system with single vacation. It follows from (2.30) that the expected waiting time in the system is given by

W_s = λE[V²]

2(λE[V ] + α₀) + λE[S²] 2(1− ρ),

which is in accordance with Takagi’s system [38].

Case 2.6: Letting p = 0, our model can be recovered to the M^[x]/G/1 queueing system with at most J vacations. Using (2.30) we have the expected waiting time in the system as:

W_s = (1− α^J0)λE[V²]

2((1− α^J0)λE[V ] + (1− α0)α^J₀) +λE[X]E[S²]

2(1− ρ) +E[X(X − 1)]E[S]

2E[X](1− ρ) , which is in accordance with Ke and Chu [28].

Case 2.7: Substituting p = 0 and J = 1 into (2.37), our system can be reduced to the

ordinary M^[x]/G/1 single vacation policy queue and it gives φ(z) = V^∗(a(z)) + α0(X(z)− 1),

which is in accordance with Choudhury’s system [5].

Case 2.8: As p = 0, our system can be simplified to the ordinary M^[x]/G/1 vacation policy queue and with at most J vacations. (2.37) can be rewritten as

φ(z) = 1− α^J0

1− α0

(V^∗(a(z))− α0) + α^J₀X(z), which is in accordance with Ke and Chu [28].

For convenient comparison of our model with the existing literature, we summarize the results listed above in Table 2.1.

2.6 Optimal Randomized Operating Policy

In this section, we develop the long-run expected cost function per unit time for the M^[x]/G/1/VAC(J) queueing system, in which p and J are the decision variables. Our objec-tive 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:

C_h≡ holding cost per unit time per customer present in the system;

C_s≡ 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 ) = C_hL_s+ Cs

E[C]

= A₁+ A₂(1− (¯pα0)^J) + A₃(1− ¯pα0) B1(1− (¯pα0)^J) + α0

(p(1− (¯pα0)^J−1) + (¯pα0)^J−1B2

), (2.46)

where

A₁ = C_h× LM^[x]/G/1, A₂ = C_hλ²E[X]E[V²]

2 ,

A₃ = C_sλ(1− ρ), B₁ = λE[V ], and

B₂ = (1− ¯pα0), with

L_M[x]/G/1 = ρ + λE[X(X− 1)]E[S] + (λE[X])²E[S²]

2(1− ρ) .

For analysis, J may be treated as a continuous variable greater than zero. Noting that 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 (2.46)) with respect to p and J , respectively, it gives

∂F (p, J )

∂p = D₁

B₁(1− (¯pα0)^J) + α₀(

p(1− (¯pα0)^J−1) + (¯pα₀)^J−1B₂), (2.47) and

∂F (p, J )

∂J =−(ln(¯pα0))(¯pα₀)^J × D₂

B1(1− (¯pα0)^J) + α0

(p(1− (¯pα0)^J−1) + (¯pα0)^J−1B2

), (2.48) where

D₁ = D₁₁× D12− D13×(

D₁₄+ D₁₅)

, (2.49)

and

D₂ = A₂B₂α₀ (

(¯pα₀)^J−1+1− (¯pα0)^J

¯ pα₀

)

+ A₂α₀p (

(1− (¯pα0)^J−1)− 1− (¯pα0)^J

¯ pα₀

)

− A3(1− ¯pα0)(

(B1− B2)− (B1 − 1)p)

, (2.50)

with

D₁₁ = B₁(1− (¯pα0)^J) + α₀(

p(1− (¯pα0)^J−1) + (¯pα₀)^J−1(1− ¯pα0)) ,

D₁₂ = B₁(1− (¯pγ0)^J) + γ₀ (

p(1− (¯pγ0)^J−1) + (¯pγ₀)^J−1(1− ¯pγ0) )

,

D₁₃ = A₂(1− (¯pα0)^J) + A₃(1− ¯pα0),

D₁₄=−B1α^J₀J ¯p^J−1+ α₀(

(1− (¯pα0)^J−1) + p(J− 1)α^J₀⁻¹p¯^J−2) , and

D₁₅= α₀(

− α₀^J−1(J− 1)¯p^J−2(1− ¯pα0) + (¯pα₀)^J−1α₀) .

For any given J , we know that:

1a. If D₁ > 0 in p ∈ [0, 1], by using (2.47) 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 (2.47) 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 arbitrary value between 0 and 1). 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 (2.48) yields ^{∂F (p,J )}_∂J > 0 which means F (p, J ) is an increasing function of J .

2b. If D₂ < 0, by using (2.48) 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 arbitrary positive integer or J =∞). 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. (2.51)

The solutions (p, J ) = (p^∗, J^∗) attain a local minimum if it satisfies the following

∂²F (p, J )

∂p² > 0, (2.52)

∂²F (p, J )

∂J² > 0, (2.53)

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. (2.54)

Noting that (2.51) 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 to search the joint suitable values 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, any positive integer).

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, any positive integer).

Case 3 If D₁ = 0 and

i. D₂ > 0 then (p^∗, J^∗) =(any value between 0 and 1, 1), ii. D₂ < 0 then (p^∗, J^∗) =(any value between 0 and 1, ∞),

iii. D2 = 0 then (p^∗, J^∗)=(any value between 0 and 1, any positive integer).

2.7 Numerical Illustration

In this section, we perform two extensive numerical examples to illustrate the joint optimum randomized behavior as discussed in Section 2.6.

According to the total expected cost function constructed in Section 2.6, we perform extensive numerical computations to illustrate the joint optimum randomized behavior and study the effect of various parameters on the optimum joint thresholds of (p, J ).

We first consider the case of D1 > 0 and D2 < 0 with the setting system’s parameters as follows:

1. p=0.5;

2. Batch size distribution of the arrival is geometric with mean E[X] = 2;

3. λ=0.6;

4. V ≡ exponential distribution with a mean E[V ] = 1;

5. S ≡ 2-stage Erlang distribution with a mean E[S] = 0.5;

6. the holding cost C_h = 10;

7. the set-up cost C_s = 1000.

The expected cost F (p, J ) for this case is shown in Figure 2.1. Noting that the minimum cost per unit time of $188.6047 is achieved at p^∗ = 0 and J^∗ is 6.

The second example is the case of D₁ > 0 and D₂ > 0 with the following system’s parameters:

1. p=0.5;

2. Batch size distribution of the arrival is geometric with mean E[X] = 3;

3. λ=0.6;

4. V ≡ exponential distribution (denoted by M with a mean E[V ] = 0.5);

5. S ≡ 2-stage Erlang distribution with a mean E[S] = 0.5;

6. the holding cost C_h = 10;

7. the set-up cost C_s= 1000.

It is seen from Figure 2.2, for the D₁ > 0 and D₂ > 0 case, that a minimum cost value per unit time of $314.8901 is achieved at p^∗ = 0 and J^∗ = 1.

The numerical results make it obvious that (i) the expected cost increases as p increases;

and (ii) J decreases as p increases. Our numerical results also agree with the conclusion in preceding section (i.e., the joint optimal values is (0, 1)-single vacation or (0, M )-multiple vacation, where M is a sufficiently large number for practice use). These special policies can be also referred to special cases 2.2 and 2.3. This implies the optimal vacation policy is exactly as single vacation or multiple vacation policy.

Table 2.1. Summary of some special cases of our model [M^[x]/G/1/VAC(J)]

Our model Under what

condition Special case Reduce to what kind of model

0

0.2

0.4

0.6

0.8 1

0 20

40 60

180 190 200 210 220 230

J p

F(p,J)

F(p^*, J^*)=188.6047 where p^* is 0 and J^* is 6.

Figure 2.1. The expected cost for the different values p and J. (D₁>0 and D₂ <0)

0

0.2

0.4

0.6

0.8 1

0 20

40 60

310 315 320 325 330 335

J p

F(p,J)

F(p^*, J^*)=314.8901 where p^* is 0 and J^* is 1.

Figure 2.2. The expected cost for different values p and J. (D₁ >0 and D₂ >0)

Chapter 3

Analysis of an Un-Reliable Server M

^[x]

/G/1 Queueing System with Randomized Vacation Policy

In Chapter 3, we extend the queueing models discussed in Chapters 2 to the M^[x]/(G1, G2)/1/VAC(J) queueing system with an un-reliable server, in which the un-reliable server operates a randomized vacation policy with at most J vacations. When the server is working, it is assumed that the server can break down at any time with a Poisson breakdown rate α.

When the server fails, it is immediately repaired in a repair facility.

This chapter is organized as follows: In Section 3.1, we describe the basic assumptions and notations of the queueing system under study. In Section 3.2, we develop the steady-state differential-difference equations for the M^[x]/(G1, G2)/1/VAC(J) queueing system by treating the elapsed service time, the elapsed repair time and the elapsed vacation time as supplementary variables. Then we solve these system equations and derive the probability generating functions of various server’s states at a random epoch. In Section 3.3, we derive the expected number of customers in the system and the expected waiting time. Section 3.4 we develop some other 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 distribution, etc. We also derive the Laplace Stieltjes Transform (LST) for the busy period and idle period, and further derive the expected length of busy period, idle period and busy cycle. In Section 3.5, we develop two main reliability indices of the presented model, namely, the steady state system availability and steady state failure frequency. In Section 3.6, we present some existing results in the literature which are special cases of our system. In Section 3.7, we develop the long-run expected cost function per unit time for the M^[x]/(G₁,G₂)/1/VAC(J) queueing system and proposed a decision criterion to 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 3.8, we perform two extensive numerical examples to illustrate the joint optimum randomized behavior as discussed in Section 3.7.

3.1 Assumptions and Notations

In this chapter, we investigate a M^[x]/(G1, G2)/1/VAC(J) queueing system with an un-reliable server, in which the server operates a randomized 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 occurring according to a compound Poisson process with arrival rate λ. Let X_k denote the number of customers belonging to the kth arrival batch, where X_k, 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. Whenever the system becomes empty, the server leaves for a vacation with random length V having distribution function V (t) and Laplace-Stieltjes transform (LST) V^∗(θ). 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 re-mains 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 server is subject to breakdowns at any time with Poisson breakdown rate α when he is working. Whenever the server fails, he is immediately repaired at a repair facility, where the repair time is an inde-pendent 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 includes both the service time of a customer and the repair time of a server. The LST of G can be expressed as follows:

G^∗(θ) =

∫ _∞

0

∑∞ n=0

e^−αt(αt)ⁿ

n! e^−θt[R^∗(θ)]ⁿdS(t),

= S^∗(

θ + α(1− R^∗(θ)))

. (3.1)

From (3.1), we obtain the first moment of G given by E[G] =− d

dθ[G^∗(θ)]|θ=0 = E[S](1 + αE[R]), (3.2)

where E[S] = −_dθ^d[S^∗(θ)]|θ=0 is the mean service time, and E[R] = −_dθ^d[R^∗(θ)]|θ=0 is the mean repair time.

In this chapter, the following notations are used.

1. λ ≡ Poisson arrival rate.

2. X_k ≡ 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. R, R(t), R^∗(θ) ≡ repair time random variable, repair time distribution function and LST of R(t), respectively.

6. V, V (t), V^∗(θ) ≡ vacation time random variable, vacation time distribution function and LST of V (t), respectively.

7. µ(x)dx ≡ the first order differential (hazard rate) functions of S.

8. ω(x)dx ≡ the first order differential (hazard rate) functions of V . 9. N (t) ≡ the number of customers in the system.

10. S⁻(t) ≡ the elapsed service time.

11. R⁻(t) ≡ the elapsed repair time.

12. V⁻(t) ≡ the elapsed time of the j^th vacation.

13. P₀(t) ≡ the probability that the server is idle but available at time t.

14. P_n(x, t)dx ≡ the probability that there are n customers in the system when the server is busy at time t.

15. Q_n(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.

16. Ωj,n(x, t)dx≡ the probability that there are n customers in the system when the server is on the j^th vacation at time t.

17. P0 ≡ the steady state probability that the server is idle but available.

18. P_n(x) ≡ the steady state probability that there are n customers in the system when the server is busy.

19. Q_n(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.

20. Ω_j,n(x)≡ the steady state probability that there are n customers in the system when the server is on the j^th vacation.

21. γ₀ ≡ the probability that there are no batches arrive during a vacation time.

3.2 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 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.

In steady state, let us assume that S(x) = 0, for x 6 0, S(∞) = 1, R(y) = 0, for y 6 0, R(∞) = 1 and V (x) = 0, for x 6 0, V (∞) = 1 and these distribution functions are continuous at x = 0, so that µ(x)dx = _1−S(x)^dS(x) , η(y)dy = _1−R(y)^dR(y) and ω(x)dx = _{1−V (x)}^{dV (x)} , where µ(x)dx can be interpreted as the conditional probability function of time for completing the service, given that the elapsed time is x. The η(y)dy and ω(x)dx can be referred to the corresponding vacation density.

The following random variables we define are used for the development of the proposed queueing system subject to server breakdowns:

N (t) ≡ the number of customers in the system, S⁻(t) ≡ the elapsed service time,

R⁻(t) ≡ the elapsed repair time, and

V_j⁻(t)≡ the elapsed time of the j^th vacation.

We introduce the following random variable for further development of the variant vacation queueing model:

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 on the 1^th vacation at time t, ...

j + 2, if the server is on the j^th vacation at time t, ...

J + 2, if the server is on the J^th vacation at time t.

The supplementary variables S⁻(t), R⁻(t) and V_j⁻(t) are introduced in order to obtain a

tri-variate Markov process {N(t), ∆(t), δ(t)}, where δ(t) = 0 if ∆(t) = 0, δ(t) = S⁻(t) if

∆(t) = 1, δ(t) = R⁻(t) if ∆(t) = 2, and δ(t) = V_j⁻(t) if ∆(t) = j + 2 (j = 1, 2, ...J ).

Furthermore, let us define the following probabilities:

P₀(t) = P_r{N(t) = 0, δ(t) = 0}, for-ward equations that govern the system under steady-state conditions can be written as follows: We solve the above equations by means of the following boundary conditions at x = 0

P_n(0) =

Ω_j,n(0) =

where a(z) = λ(1− X(z)). Similarly proceeding in the usual manner with (3.5)-(3.8), we have

where x > 0.

Solving the partial differential equations (3.13)-(3.15), we obtain

P (x; z) = P (0; z)[1− S(x)]e^−A(z)x, (3.17)

Q(x, y; z) = Q(x, 0; z)[1− R(y)]e^−a(z)y, (3.18)

and

Ω_j(x; z) = Ω_j(0; z)[1− V (x)]e^−a(z)x, j = 1, 2, ...J, (3.19) where A(z) = a(z) + α(

1− R^∗(a(z))) .

Solving the differential equation (3.6) yields

Ω_j,0(x) = Ω_j,0(0)[1− V (x)]e^−λx, j = 1, 2, ...J. (3.20) Now, (3.20) is multiplied by ω(x) on both sides for j = J and integrating with x from 0 to

∞, we then have

∫ _∞

0

Ω_J,0(x)ω(x)dx = ΩJ,0(0)γ₀, (3.21)

where γ₀ = V^∗(λ). Inserting (3.21) in (3.10), we can recursively obtain Ω_j,0(0) = Ω_J,0(0)

(¯pγ₀)^J−j, j = 1, 2, ..., J − 1. (3.22)

Substituting (3.22) into (3.3) and after some algebraic manipulation, we have Ω_J,0(0) = λP₀

γ₀[

1 + _{( ¯pγ}^{p(1−(¯pγ0)J−1)}

0)^J−1(1−¯pγ⁰)

]. (3.23)

From (3.22) and (3.23) we finally obtain Ω_j(0; z) = Ω_j,0(0) = λP₀

(¯pγ₀)^J−jγ₀[

1 + _{( ¯pγ}^{p(1−(¯pγ0)J−1)}

0)^J−1(1−¯pγ0))

], j = 1, 2, ..., J. (3.24)

Integrating (3.20) with respect to x from 0 to ∞ and using the well-known result of renewal theory

∫ _∞

0

e^−λx(1− V (x))dx = 1− γ0

λ ,

we obtain

From (3.24) and (3.25), it finally yields Ω_j,0= P₀(1− γ0)

(¯pγ₀)^J−jγ₀[

1 + _{( ¯pγ}^{p(1−(¯pγ0)J−1)}

0)^J−1(1−¯pγ⁰)

], j = 1, 2, ..., J. (3.26)

Noting that Ω_j,0 represents the steady-state probability that there are no customers in the system when the server is on the j^th 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 (3.17), (3.19) and (3.24) into (3.16) we get on simplification

P (0; z) = λP₀(1− (¯pγ0)^J)V^∗(a(z))

Solving P (0; z) from (3.28) and using (3.24) yields

P (0; z) =

It follows from (3.17) and (3.29) that

P (x; z) =

Inserting (3.18) with the boundary condition (3.11), Q(x, 0; z) can be expressed as

Q(x, 0; z) = αP (x; z). (3.32)

Utilizing (3.17) and (3.32) in (3.18), we get

Q(x, y; z) = αP (0; z)[1− S(x)]e^−A(z)x[1− R(y)]e^−a(z)y. (3.33) Inserting (3.29) into (3.33), we obtain

Q(x, y; z) =

Calculating the double integral∫_∞

0

∫_∞

0 Q(x, y; z)dxdy, we finally obtain

Q(z) =

Using (3.19) and (3.24) we have Ω_j(z) = P₀(V^∗(a(z))− 1)

[X(z)− 1](¯pγ0)^J−jγ0

[1 + _{( ¯pγ}^{p(1−(¯pγ0)J−1)}

0)^J−1(1−¯pγ0)

], j = 1, 2, 3, ...J. (3.36)

The unknown constant P0 can be determined by using the normalization condition (3.12), which is equivalent to P₀+ P (1) + Q(1) +∑J

Noting that (3.37) represents the steady state probability that the server is idle but available in the system. Also from (3.37), we have ρ_H < 1 which is the necessary and sufficient condition under which steady state solution exists.

Let Φ(z) = P₀ + P (z) + Q(z) +∑J

j=1Ω_j(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)

3.3 The Expected Number of Customers in the System and the Expected Waiting Time

In (3.38), we evaluate _dz^dΦ(z)|z=1 by using L’hopital rule which leads to the expected number of customers, L_s, in the system given by

L_s= ρ_H + λE[X(X− 1)]E[G]

By using Little’s formula, we obtain the expected waiting time in the queue, W_q, given by

3.4 Other System Characteristics

In this section, we first develop the queue size distribution at a departure epoch in subsection 3.4.1. In subsection 3.4.2 we develop the system size distribution at busy period initiation epoch. Finally, we derive the busy period and idle period distributions in subsection 3.4.3.

3.4.1 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 M^[x]/(G₁,G₂)/1/VAC(J) queueing system. 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. This leads to the following

Φ⁺_l = K₀

∫ _∞

0

µ(x)P_l+1(x)dx, l = 0, 1, ... (3.41)

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 implies the probability generating function of the departure point queue size distribu-tion as

From (3.44), we observe that Φ⁺(z) can be decomposed into two independent terms:

Φ⁺(z) = 1− X(z)

E[X](1− z)× Φ(z). (3.45)

It should be noted that the departure point queue size distribution given by (3.45) can be decomposed into two independent random variables: One (the first term) is the number of customers placed before a tagged customer in a batch in which the tagged customer arrives and the other (the second term) is the stationary system size of the M^[x]/(G₁,G₂)/1/VAC(J) queueing system.

3.4.2 System Size Distribution at Busy Period Initiation Epoch

We first define φ_n(n = 1, 2, ...) as the steady-state probability that an arbitrary (tagged) customer finds n customers in the system at the busy initiation epoch (or completion epoch of the idle period). This implies that t_l (l = 0, 1, 2, ...) are the initiation epochs of the busy period and N (t_l) is the number of customers in the system at the time instant t_l, then we have

φ_n = lim_l→∞Pr(N (t_l) = n), n = 1, 2, ....

Conditioning on the number of customers which arrive during the first vacation and from the concept of Wolff [49], we have the following steady-state equation

φ_n =( to be 1, and γ_k = Pr(k batches arrive during a vacation time).

Now multiplying (3.46) by appropriate powers of z and then taking summation over all possible values of n, we get the p.g.f. of {φn} given by

Noting that (3.47) represents the p.g.f. of the number of customers in the system at the completion epoch of the idle period and this is equivalent to the p.g.f. of the system size distribution at busy period initiation epoch.

3.4.3 The Expected Length of the Completion Period and Idle Period

Let H^∗(θ) and I^∗(θ) represent the LST of the completion period (including busy period and breakdown period) and idle period for the M^[x]/(G₁,G₂)/1/VAC(J) queueing system.

Utilizing the arguments by Takagi [38] and Tang [44], H^∗(θ) and I^∗(θ) can be expressed as H^∗(θ) = 1− (¯pγ0)^J M^[x]/G/1 queueing model with an un-reliable server.

Now, we further define the following

E[H]≡ the expected length of completion period, E[I]≡ the expected length of idle period,

E[C]≡ the expected length of busy cycle.

Employing (3.49) and (3.50), we obtain E[H] =

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

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

在文檔中 具隨機假期策略之M[x]/G/1排隊系統分析 (頁 32-0)