Organization

This dissertation is organized as follows. In Chapter 1, we review some existing works concerning vacation queueing models and point out our research objectives.

In Chapter 2, we study the M^[x]/G/1 queueing system with a reliable server in which the server operates a randomized vacation policy with at most J vacations. We first develop the differential-difference equations governing the variant vacation system and derive the probability generating functions (p.g.f) of system size in various server states. Secondly, we also derive some other important system characteristics such as the expected number of customers in the system, the expected waiting time, the queue size distribution at a departure epoch, the system size distribution at busy period initiation epoch, the system size distribution due to idle period and the distributions of idle period and busy period.

Thirdly, we demonstrate some existing results in the literature which are special cases of our system. Fourthly, a long-run expected cost function per unit time is constructed to determine the optimal control policy. Finally, we provide a decision criterion to find the joint suitable values (p^∗, J^∗), some numerical examples are presented for illustrative purpose.

In Chapter 3, we investigate the M^[x]/G/1 queueing system with an un-reliable server.

The server operates a randomized vacation policy with at most J vacations. When the server breaks down, it is immediately repaired in a repair facility. For such a system, we first develop the probability generating functions (p.g.f) of the number of customers present in the system. Secondly, we also derive other system characteristics such as the expected number of customers in the system, the expected waiting time, the system size distribution at busy period initiation epoch, the queue size distribution at a departure epoch and the busy and idle period distributions. Thirdly, some important reliability indices of our system are derived. Fourthly, we show some existing results in the literature which are special cases of our system. Fifthly, a long-run expected cost function per unit time is constructed to determine the optimal control policy. Finally, we provide a decision criterion to find the

suitable values (p^∗, J^∗), some numerical examples are performed for illustrative purpose.

In Chapter 4, we extend the system considered in Chapters 2 and 3 to the M^[x]/G/1 queueing system with an un-reliable server and delayed repair under randomized vacation policy with at most J vacations. For such a system, we first develop the probability gen-erating function (p.g.f) of the number of customers present in the system. Secondly, we also develop other system characteristics such as the system size distribution at busy period initiation epoch, the queue size distribution at a departure epoch and the busy and idle period distributions. Thirdly, some important reliability indices of our system are derived.

Fourthly, we present some existing results in the literature which are special cases of our system. Fifthly, a long-run expected cost function per unit time is constructed to determine the optimal control policy. Finally, we develop an efficient decision criterion for searching the joint suitable values (p^∗, J^∗). Some examples are performed numerically to illustrate the optimization approach.

In Chapter 5, we summarized the results obtained in our investigation and give some conclusions, some recommendations for the future research are also given.

Chapter 2

Analysis of a Reliable Server M

^[x]

/G/1 Queueing System with Randomized Vacation Policy

In this chapter, we investigate the randomized vacation policy for the M^[x]/G/1 queue-ing system with a reliable server in which the server operates a randomized vacation policy with at most J vacations. The server leaves for a vacation when the system becomes empty.

When the server returns from the vacation, he applies a randomized vacation policy and decides to take another vacation, to remain dormant in the system or to provide service for the waiting customers. The detail descriptions of randomized vacation policy queueing model are stated in Section 1.1.

This chapter is organized as follows: In Section 2.1, some basic assumptions and notations of the queueing system under study are given. Section 2.2 we develop the steady-state differential-difference equations for the M^[x]/G/1/VAC(J) queueing system by treating the elapsed service time 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 2.3, using the p.g.f’s obtained in Section 2.2, we obtain the expected number of customers in the system and the expected waiting time for the proposed queueing system. Section 2.4 we develop some other important system characteristics such as the the queue size distribution at a departure epoch, the system size distribution at busy period initiation epoch, the system size distribution due to idle period including vacation and the busy period and idle period distribution, etc. We also propose 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 2.5, we demonstrate some existing results in the literature which are special cases of our system. In Section 2.6, we develop the long-run expected cost function per unit time for the proposed queueing system and propose a criterion to determine the suitable values of the control variables p and J , say p^∗ and J^∗, so as to minimize the cost function. Some numerical examples are presented in Section 2.7 for illustrative purpose.

2.1 Assumptions and Notations

In this chapter, we investigate an M^[x]/G/1 system in which the reliable 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 discussed in this chapter 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(X_k = n) = χ_n, n = 1, 2, 3, ...

The service time provided by a single server is an independent and identically distributed 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 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 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. It is assumed that various stochastic processes involved in the system are independent of each other.

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. S, S(t), S^∗(θ) ≡ service time random variable, service time distribution function and LST of S(t), respectively.

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

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

6. ω(x)dx ≡ the first order differential (hazard rate) functions of V . 7. Q(t) ≡ the number of customers in the system.

8. S⁻(t) ≡ the elapsed service time.

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

10. R₀(t) ≡ the probability that the server is idle but available at time t.

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

12. 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 given that the elapsed vacation time is x.

13. R0 ≡ the steady state probability that the server is idle but available.

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

15. 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 at time.

16. α₀ ≡ the probability that there are no batches arrive during a vacation time.

2.2 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 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, V (x) = 0, for x 6 0, V (∞) = 1 and these two distribution functions are continuous at x = 0, so that µ(x)dx = _1−S(x)^dS(x) and ω(x)dx = _{1−V (x)}^{dV (x)} , where µ(x)dx and ω(x)dx are the first order differential (hazard rate) functions of S and V respectively.

We define the state of the system at time t as follows:

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

and

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

The following random variables we define are used for the development of M^[x]/G/1/VAC(J) queueing system:

0, if the server is idle in the system at time t, 1, if the server is busy at time t,

The supplementary variables S⁻(t) and V_j⁻(t) are introduced in order to obtain a tri-variate Markov process {Q(t), ∆(t), δ(t)}, where δ(t) = 0 if ∆(t) = 0, δ(t) = S⁻(t) if ∆(t) = 1, and δ(t) = V_j⁻(t) if ∆(t) = j + 1 (j = 1, 2, ..., J ).

Furthermore, let us define the following probabilities:

R₀(t) = P_r{Q(t) = 0, δ(t) = 0},

Pn(x, t)dx = Pr{Q(t) = n, δ(t) = S⁻(t); x < S⁻(t)≤ x + dx}, x > 0, n ≥ 1,

Ω_j,n(x, t)dx = P_r{Q(t) = n, δ(t) = V⁻(t); x < V_j⁻(t)≤ x+dx}, x > 0, n ≥ 0, 1 ≤ j ≤ J.

In steady-state, we can set R₀ = lim_t→∞R₀(t) and limiting densities P_n(x) = lim_t→∞P_n(x, t) and Ω_j,n(x) = lim_t→∞Ω_j,n(x, t). According to Cox [10], the steady-state Kolmogorov forward equations that govern the system can be written as follows:

λR₀ =

We solve the above equations by means of the following boundary conditions at x = 0

P_n(0) =

Ω_j(x; z) =

∑∞ n=0

zⁿΩ_j,n(x), |z| 6 1, 1 6 j 6 J.

Now multiplying (2.2) by zⁿ (n = 1, 2, 3, ...) and then adding the equations up term by term, it gives

∂P (x; z)

∂x + [a(z) + µ(x)]P (x; z) = 0, (2.9)

where a(z) = λ(1− X(z)). Similar proceeding in the usual manner with (2.3)-(2.5), we have

∂Ω_j(x; z)

∂x + [a(z) + ω(x)]Ω_j(x; z) = 0, (2.10)

and

P (0; z) =

∑J j=1

∫ _∞

0

Ω_j(x; z)ω(x)dx + 1 z

∫ _∞

0

P (x; z)µ(x)dx + λX(z)R₀−

∑J j=1

Ω_j(0; z)

− λR0, (2.11)

where x > 0.

Solving the partial differential equations (2.9) and (2.10), we obtain

P (x; z) = P (0; z)[1− S(x)]e^−a(z)x, (2.12)

and

Ω_j(x; z) = Ω_j(0; z)[1− V (x)]e^−a(z)x, j = 1, 2, ..., J. (2.13) Solving the differential equation (2.3) yields

Ωj,0(x) = Ωj,0(0)(1− V (x))e^−λx, j = 1, 2, ..., J. (2.14) Now (2.14) is multiplied by ω(x) on both sides for and integrating with x from 0 to ∞, we then have

∫ _∞

0

Ω_j,0(x)ω(x)dx = Ω_j,0(0)α₀, (2.15)

where α₀ = V^∗(λ). Inserting (2.15) in (2.7), we can recursively obtain Ω_j,0(0) = Ω_J,0(0)

(¯pα₀)^J−j, j = 1, 2, ..., J − 1. (2.16)

Substituting (2.15) and (2.16) into (2.1) and after some algebraic manipulation, we have

From (2.16) and (2.17) we finally obtain Ω_j(0; z) = Ω_j,0(0) = λR0

(¯pα₀)^J−jα₀[

1 + _{( ¯pα}^{p(1−(¯pα0)J−1)}

0)^J−1(1−¯pα0))

], j = 1, 2, ..., J. (2.18)

Integrating (2.14) with respect to x from 0 to∞ we have Ω_j,0= Ω_j,0(0)

∫ _∞

0

[1− V (x)]e^−λxdx = 1

λΩ_j,0(0)(1− α0). (2.19)

From (2.18) and (2.19), it finally yields Ωj,0= R₀(1− α0)

(¯pα0)^J−jα0

[1 + _{( ¯pα}^{p(1−(¯pα0)J−1)}

0)^J−1(1−¯pα0)

], j = 1, 2, ..., J. (2.20)

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 (2.12), (2.13) and (2.18) into (2.11) we get on simplification

P (0; z) = λR₀(1− (¯pα0)^J)V^∗(a(z))

Solving P (0; z) from (2.22) and using (2.18) yields

P (0; z) =

It follows from (2.12) and (2.23) that

P (x; z) =

which leads to Using (2.13), (2.18) and the well-known result of renewal theory

∫ _∞

The unknown constant R₀ can be determined by using the normalization condition (2.8), which is equivalent to R₀+ P (1) +∑J

Noting that (2.27) represents the steady state probability that the server is idle but available in the system. Also from (2.27), we have ρ < 1 which is the necessary and sufficient condition under which steady state solution exists.

Let Φ(z) = R₀+P (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− ρ)S^∗(a(z))(z− 1)

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

In (2.28), 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 + λE[X(X− 1)]E[S] + (λE[X])²E[S²] 2(1− ρ)

+ λ²E[X](1− (¯pα0)^J)E[V²] 2

(

λE[V ](1− (¯pα0)^J) + α₀[

(¯pα₀)^J−1(1− ¯pα0) + p(1− (¯pα0)^J−1)]). (2.29) Noting that, the first and second terms in (2.29) represent the expected number of customers in the system for the ordinary M^[x]/G/1 queueing system.

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

W_q = E[X(X− 1)]E[S] + λ(E[X])²E[S²] 2E[X](1− ρ)

+ λ(1− (¯pα0)^J)E[V²]

2 (

λE[V ](1− (¯pα0)^J) + α₀[

(¯pα₀)^J−1(1− ¯pα0) + p(1− (¯pα0)^J−1)]). (2.30)

2.4 Other System Characteristics

In this section, we first develop the queue size distribution at a departure epoch in subsection 2.4.1. In subsection 2.4.2 we develop the system size distribution at busy period initiation epoch. In subsection 2.4.3 we derive the system size distribution due to idle period including vacation. Finally we derive the busy period and idle period distributions in subsection 2.4.4.

2.4.1 Queue Size Distribution at a Departure Epoch

We derive the probability generating function of queue size distribution 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 = C₀

∫ _∞

0

µ(x)P_l+1(x)dx, l = 0, 1, ... (2.31)

where Φ⁺_l = Pr{A departing customer will see l customers in the system}, and C0 is the normalizing constant.

Using the normalization condition Φ⁺(1) = 1, it gives

C₀ = 1− ρ which leads to the probability generating function of the departure point queue size distri-bution as

From (2.34), one see that Φ⁺(z) can be decomposed into two independent terms:

Φ⁺(z) = 1− X(z)

E[X](1− z)× Φ(z). (2.35)

It should be noted that the departure point queue size distribution given by (2.35) 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]/G/1/VAC(J) queueing system.

2.4.2 System Size Distribution at Busy Period Initiation Epoch

First, we define φn (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 Q(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 α_k = Pr(k batches arrive during a vacation time).

Now multiplying (2.36) 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 (2.37) 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.

2.4.3 System Size Distribution Due to Idle Period

Let us define ξn (n = 0, 1, 2...) as the probability that a batch of n customers arrived before a tagged customer during the forward recurrence time (residual life) of the idle period where the tagged customer arrived. The batch of arriving customers associated with the tagged customer is randomly chosen from the arriving batch that occurs at the completion

epoch of the idle period (busy period initiation epoch). Following arguments of Burke [4]

and applying renewal theory, we obtain the p.g.f of the number of customers that arrive during the residual life of the idle period given by

ξ(z) =

(1− φ(z))

(1− z)E[φ]. (2.39)

Using (2.37) in (2.39), ξ(z) can be expressed as

ξ(z) = 1− ¯pα0− (1 − (¯pα0)^J)(V^∗(a(z))− α0)− (1 − ¯pα0)[_p(α0−¯p^J−1α^J₀) Noting that (2.40) is the p.g.f. of the number of customers that arrive during a time interval from the beginning of the idle period to a random point in the idle period. We may view it as the system size distribution due to the idle period including vacation times.

2.4.4 The Expected Length of the Busy Period and Idle Period

Let B^∗(θ) and I^∗(θ) represent the LST of the busy period and idle period for the M^[x]/G/1/VAC(J) queueing system. Utilizing the arguments by Takagi [38] and system definition, B^∗(θ) and I^∗(θ) can be expressed as customer in the ordinary M^[x]/G/1 queueing model.

Now, we further define the following

E[B]≡ the expected length of busy period,

E[I]≡ the expected length of idle period, E[C]≡ the expected length of busy cycle.

Employing (2.41) and (2.42), we obtain E[B] =

2.5 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^∗:

• The criterion to search the joint suitable values p^∗ and J^∗:

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