Sensitivity Analysis - Optimization of the N Policy M/G/1 Queue with Server Breakdowns and

Chapter 2 Optimization of the N Policy M/G/1 Queue with Server Breakdowns and

2.5 Sensitivity Analysis

positive integer value of

is one of the integers surrounding



N

F N ( ) N

N N

2.5 Sensitivity Analysis

A system analyst often concerns with how the system performance measures can be affected by the changes of the input parameters in the investigated queueing service model. Sensitivity investigation on the queueing model with critical input parameters may provide some answers to this question. In the following, we conduct some sensitivity investigations on the optimal value based on changes in the values of the system parameters

N

, , , ,

λ μ α β γ and cost parameters

C C C C , , ,

From (2.26), we perform some algebraic manipulations with respect to system

Differentiating (2.27) and (2.28) with respect to

λ again and substituting (2.29) and

(2.30) into the resulting differentiation from (2.27) and (2.28), respectively, we have

The above results show that the graph of is concave downward with respect to

N

λ

, which attains its maximum value under two different parameter settings satisfying (2.29) and (2.30), respectively. Differentiating

N

^* with respect to μ yields

where

ρ λμ

= _S =

λ μ

. Thus

N

^* is increasing in μ . Similarly, differentiating

σ_U2 is a function of the parameter γ . For example, suppose the startup time distribution obeys the Erlang-k (k > 1) stage distribution with mean

μ

_U (=1

γ

Substituting σ_U²

=

μ_U²

k into (2.26) and then differentiating N

^* with respect to

Two situations are considered while investigating the behavior of

∂ N

∂

μ_U : Case (i): If θ₂²

− 4

λθ₃

> 0

and setting

∂ N

∂

μ_U = 0, then we obtain again and using (2.37), we finally get

( )

implies that

N

^* is also a concave downward function of γ . Therefore, from (2.37), we may obtain

γ . On the other hand, it can easily see from (2.26) that is increasing in

N

* s

,

C C

and decreasing in

C C

_h, _i.

2.6 Numerical Computations

We present some numerical computations to demonstrate the analytical results obtained, and show how to make the decision based on minimizing the cost function (see (2.25)). Since the cost function is only related to system parameters

λ

, μ ,

α

, β , γ in which

μ

_S =1

μ

_R =1

β

μ

_U =1

γ

and σ is a function of γ , then _U² (2.25) is independent of service time distribution and repair time distribution except for startup time distribution. The sensitivity investigation focuses on the Erlang-2 startup time distribution. First, we fix the following cost parameters

C =1000,

C

_h=5,

C

sp= 100,

C

_i=60 and consider the following five cases.

Case 1: We select μ =0.5, 1, 1.5, 2,

α

=0.05, β =3, γ =3, and vary the values of

λ

Case 2: We select

λ

=0.2, 0.4, 0.6, 0.8,

α

= 0.05, β =3, γ =3, and vary the values of μ .

We observe from Figure 2.1 that (i) the local maximum value of is moving from left to right as

N

μ increases; and (ii) as

λ

is fixed,

N

^* is getting larger as μ increases. From Figure 2.2, we see that (i)

N

^* increases in μ ; (ii) if μ is small enough,

N

^* increases quickly; (iii) if μ is large and

ρ λ μ

is small enough,

N

* is insensitive; and (iv) if μ is fixed and large enough,

N

^* increases in

λ

. Numerical results of Case 1 and Case 2 are provided in Table 2.1.

Case 3: We select

λ

= 0.5, μ = 1, β = 1, 2, 3, 4, γ = 3 and vary the values of

α

. Case 4: We select

λ

= 0.5, μ = 1,

α

= 0.4, 0.8, 1.2, 1.6, γ = 3 and vary the values

of β .

We observe from Figure 2.3 that (i)

N

^* decreases in

α

. As

α

is fixed, the larger β has larger

N

^*; (ii)

N

^* has an upper bound as

α

closes to zero, and (iii)

is not insensitive to

N

α

. It can easily observe from Figure 4 that (i)

(see Figure 2.6). Figure 2.5 and Figure 2.6 show that may be too insensitive to changes in

N

γ as γ is greater than 0.4. The numerical results are presented in Table 2.3. other hand, we select

C

C =1000,

C

_h=5 and change the specified values of (

C

_sp, ).

Table 2.5 reveals that is insensitive to (

C

N

C

_sp,

C

_i).

Finally, we make comparisons between our model and existing literature (see Pearn et al. [35]). According to the parameters setting by [35], we perform a numerical experiment based on

λ

=0.4,

μ

_S =1,

σ

_S=1,

α

=0.05,

μ

_R=0.2,

σ

_R=1,

=5, =50, =10, =100 and

C

C =200. In addition, we fix startup cost

C

_sp

= 90 and vary the parameter value (γ ) of exponential startup distribution from 0.1 to 1 and

N

from 1 to 25. Figure 2.7 shows that our model approaches to that by [35] as

γ tends to large enough (

μ

tends to small enough).

Figure 2.1 Plots of (

λ

, N^*) with μ =0.5, 1.0, 1.5, 2.0,

α

=0.05, β =3, γ =3,

C =1000,

C

_h=5,

C

_sp= 100,

C

_i=60

N

Figure 2.2 Plots of (

μ

, N^*) with

λ

=0.2, 0.4, 0.6, 0.8,

α

=0.05, β =3, γ =3,

C =1000,

C

_h=5,

C

_sp= 100,

C

_i=60

N

Table 2.1 The optimal and minimum expected cost with various values of

N

F N )

( ^*

( , )

λ μ .

α =0.05,

β

=3,

γ

=3, C_s =1000, C_h =5, C_sp =100,C_i =60 ( ,λ μ)

(0.3, 0.5) (0.3, 1.0) (0.3, 1.5) (0.3, 2.0) (0.2, 1.0) (0.4, 1.0) (0.6, 1.0) (0.8, 1.0)

N* 7 9 10 10 8 10 10 8

( *)

F N 55.3856 85.1964 94.5549 99.1349 85.5034 82.2018 69.7010 47.7634

Figure 2.3 Plots of (

α

, N^*) with

λ

=0.5, μ =1, β =1, 2, 3, 4, γ =3,

C =1000,

h=5,

C C

_sp= 100,

C

_i=60

N

Figure 2.4 Plots of (

β

, N^*) with

λ

=0.5, μ =1,

α

=0.4, 0.8, 1.2, 1.6, γ =3,

C =1000,

C

_h=5,

C

_sp= 100,

C

_i=60

N

Table 2.2 The optimal and minimum expected cost with various values of

N

F N )

( ^*

( , )

α β .

λ=0.5,

μ

=1,

γ

=3, C_s =1000, C_h =5, C_sp=100, C_i =60

( ,α β) (0.5, 1.0) (0.5, 2.0) (0.5, 3.0) (0.5, 4.0) (0.4, 2.0) (0.8, 2.0) (1.2, 2.0) (1.6, 2.0)

N* 7 9 9 9 9 8 6 4

( *)

F N 48.6806 63.8626 68.4962 70.8162 66.7410 54.7399 41.5367 26.2598

Figure 2.5 Plots of (

γ

, N^*) with

λ

=0.3, μ =1,

α

=0.05, β =3,

C =1000,

C

_h=5,

C

sp= 100,

C

_i=60

N

Figure 2.6 Plots of (

γ

, N^*) with

λ

=0.3, μ =1,

α

=0.05, β =3,

C =500,

C

_h=5,

C

sp= 100,

C

_i=40

N

Table 2.3 The optimal and minimum expected cost with various values of

N

F N )

( ^*

γ .

λ=0.3,

μ

=1, α =0.05,

β

=3, C_s =1000, C_h =5, C_sp =100, C_i =60

γ 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2

N* 9 9 9 9 9 9 9 9

( *)

F N 85.4241 85.2503 85.2181 85.2068 85.2016 85.1987 85.1970 85.1959

Table 2.4 The optimal and minimum expected cost with various values of

N

F N )

( ^*

(C C_s, _h).

λ=0.3,

μ

=1, α =0.05,

β

=3,

γ

=3, C_sp=100, C_i =60

(C C_s, _h) _{(1000, 5)} (1000, 10) (1000, 15) (1000, 20) (400, 10) (600,10) (800, 10) (900,10)

N* 9 6 5 5 4 5 6 6

( *)

F N 85.1964 101.9221 114.0319 124.3333 78.3476 87.3775 95.0861 98.5041

Table 2.5 The optimal and minimum expected cost with various values of

N

F N )

( ^*

. (C_sp,C_i)

λ=0.3,

μ

=1, α =0.05,

β

=3,

γ

=3, C_s =1000, C_h =5

(C_sp,C_i) _{(80, 20)} _{(80, 30)} _{(80, 40)} _{(80, 50)} _{(35, 25)} _{(45, 25)} _{(55 , 25)} _{(65, 25)}

N* 9 9 9 9 9 9 9 9

( *)

F N 57.5492 64.4228 71.2964 78.1701 60.6423 60.7187 60.7951 60.8714

Figure 2.7 The total expected cost

F N

_O( ) for different values of γ and .

N

) (N F

N

Chapter 3 Maximum Entropy Analysis to the N Policy M/G/1 Queue with Server Breakdowns and Startup Times

We study a single removable and unreliable server in the policy M/G/1 queue with general startup times where arrivals form a Poisson process and service times are generally distributed. When customers are accumulated in the system, the server is immediately turned on but is temporarily unavailable to the waiting customers. He needs a startup time before providing service until the system becomes empty. The server is subject to breakdowns according to a Poisson process and his repair time obeys an arbitrary distribution. We use the maximum entropy principle to derive the approximate formulas for the steady-state probability distributions of the queue length. We perform a comparative analysis between the approximate results with established exact results for various distributions, such as exponential (M), k-stage Erlang (E

N

k), and deterministic (D). We demonstrate that the maximum entropy approach is accurate enough for practical purposes and is a useful method for solving complex queueing systems.

3.1 Assumptions and Notations

It is assumed that customers arrive according to a Poisson process with parameter λ and service times are independent and identically distributed (i.i.d.) random variables having a general distribution function, ( ) with a mean service time

S( )

F t t ≥ 0

μ

_S and a finite variance

σ

_S². The server is subject to breakdowns at any time with Poisson breakdown rate α when he is turned on and working. When the server fails, he is immediately repaired at a repair facility, where the repair times are independent and identically distributed random variables obeying a general distribution function

F t

_R( )(

t ≥ 0

) with a mean repair time

μ

_R and a finite variance

σ

R. Arriving-customers form a single waiting line based on the first-come, first- served (FCFS) discipline. The server can serve only one customer at a time and the service is independent of the arrival of the customers. 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 the server, customers continue to arrive following a Poisson process. Furthermore, when the number of customers in

the system reaches a specific level, denoted by , the server is immediately turned on (i.e. begin startup) but is temporarily unavailable to the waiting customers. He requires a startup time with random length before starting service. The startup times are independent and identically distributed random variables obeying a general distribution function ( ) with a mean startup time

N

U( )

F t t ≥ 0 μ

_U and a finite

variance

σ

_U² . Once the startup is over, the server begins serving the waiting customers until there are no customers in the system. Service is allowed to be interrupted if the server breaks down, and the server is immediately repaired. Once the server is repaired, he immediately returns to serve customers until the system becomes empty.

The following notations and probabilities are used throughout this chapter.

N −

threshold with server breakdowns and general startup times

N

( )

W z

− p.g.f. of the number of customers that arrive during the turned-off plus the startup period

U( )

f

⋅ − Laplace-Stieltjes transform (LST) of startup time

L

N − expected number of customers in the policy M/G/1 queue with server breakdowns and general startup times

N

0,_I( )

P n

− probability that there are customers in the system when the

n

server is turned off, where

n

=0, 1, 2,",

N

−1

0,_U( )

P n

− probability that there are customers in the system when the server is startup, where

n

, 1, +2,

n

N N

N

1( )

P n

− probability that there are customers in the system when the server is turned on and working, where

n

server is in operation but found to be broken down, where

n

U

r− remaining startup time for the server begin startup

W

−

approximate waiting time in the queue

3.2 The Expected Number of Customers in the System

Let denote the probability generating function (p.g.f.) of the number of customers in the ordinary M/G/1 queue with reliable server. From Kleinrock [28, p.

194], we have

Let be a random variable representing the completion time of a customer, which includes both the service time of a customer and the repair time of a server.

Applying the well-known formula for the p.g.f. of the number of customers in the ordinary M/G/1 queue with reliable server, the p.g.f. of the number of customers in the ordinary M/G/1 queue with unreliable server is given by

H

(1 )(1 ) ( ) completion time. Note that

ρ

_H is traffic intensity and it should be assumed to be less than unity. It should be noted that expression (3.2) is obtained only by replacing service times by completion times in the formula of the ordinary M/G/1 queue with a reliable server.

We consider that the server is on “extended vacation” during the turned-off period

I and startup period

U

_N, the lengths of which equal (

I

_N +

U ). Following

_N the result of Medhi and Templeton [34], we obtain

[

¹ ^{( )}

]

⁽ system with server breakdowns and general startup times;

N

( )

W z

≡ the p.g.f. of the number of customers that arrive during a vacation of length

I

_N +

U ;

≡ [the p.g.f. of the number of customers that arrive during

I ]

_N ×[the p.g.f. of the number of customers that arrive during

U

_N];

≡

z f

^N _U(

λ λ

−

z)

, where

f

_U ( )⋅ is the LST of startup time. with server breakdowns and general startup times. Thus we have

L

N

( )

N N z

L = G ′ z

₌

2 2

3.3 The Maximum Entropy Results

In this section, we will develop the maximum entropy solutions for the steady-state probabilities of the policy M/G/1 queue with server breakdowns and general startup times. Let us define

N

0,_I( )≡

P n

probability that there are customers in the system when the server is

turned off, where .

P n

probability that there are customers in the system when the server is

startup, where .

P n

probability that there are customers in the system when the server is turned on and working, where

n

1, 2, 3,

= "

n

2( )≡

P n

probability that there are customers in the system when the server is in operation but found to be broken down, where

n

1, 2, 3,

= "

n

In order to derive the steady-state probabilities , and (

i

= 1, 2) by using the maximum entropy principle, we formulate the maximum entropy model in the following. Following El-Affendi and Kouvatsos [12], the entropy function can be illustrated mathematically as

0,_I( )

There are five basic known results from the literature (see [8] and [54]) that facilitate the application of the maximum entropy formalism to study the

N

policy

M/G/1 queue with server breakdowns and general startup times. The maximum entropy solutions are obtained by maximizing (3.5) subject to the following five constraints, written as,

(i) normalizing condition

, (3.6)

(ii) the probability that the server is startup

[ ]

(iii) the probability that the server is busy

(iv) the probability that the server is broken down

(v) the expected number of customers in the system

, (3.10)

2 0, 1 3 1 4 2

where

τ

₁ to

τ

₅ are the Lagrangian multipliers corresponding to constraints (3.6)-(3.10), respectively.

3.3.1 The maximum entropy solutions

To get the maximum entropy solutions, , , , maximizing in (3.5) subject to constraints (3.6)-(3.10) is equivalent to maximizing (3.12).

0,_U( )

P n P n

₁( )

P n

₂( )

The maximum entropy solutions are obtained by taking the partial derivatives of

y with respect to

, and (

i = 1, 2), and setting the results

equal to zero, namely,

0,_I( )

It implies from (3.13)-(3.16) that we obtain

1 5

(3.17)-(3.20) in terms

φ φ φ φ

₁, , , ₂ ₃ ₄ and

φ

₅ given by

Substituting (3.22)-(3.24) into (3.7)-(3.9), respectively, yields

Substituting (3.11) and (3.22)-(3.24) into (3.10) and doing the algebraic manipulations, we obtain

3.4 The Exact and Approximate Expected Waiting Time in the Queue

In this section, we develop the exact and the approximate formulae for the expected waiting time in the policy M/G/1 queue with server breakdowns and general startup times as follows.

N

3.4.1 The exact expected waiting time in the queue

Let denote the exact expected waiting time in the queue. Using (3.4) and Little’s formula, we obtain

W

3.4.2 The approximate expected waiting time in the queue

We define the idle state, the startup state, the busy state, and the repair state as follows:

(i) Idle state denoted by

I : the server is turned off and the number of customers

_S waiting in the system is less than or equal to

N − 1

(ii) Startup state denoted by : the server begins startup and the number of customers waiting in the system is greater than or equal to .

U

N

(iii) Busy state denoted by

B : the server is busy and provides service to a customer.

_S (iv) Repair state denoted by

R : the server is broken down and being repaired.

Following Borthakur et al. [8], we find the expected waiting time of customer at the states

C I ,

U

_S,

B and

R as follows. Suppose that a customer

_S finds customers waiting in the queue for service in front of him, while the system is at any one of the states

C n

I ,

U

_S,

B and

R are described, respectively, as follows:

_S (i) In idle state

I : The server will begin startup after (

N

− − customers arrive

n

in the system. Thus customer

C

will be served until (N− − customers

n 1)

arrive and customers in front of him waiting for service. The expected waiting time of customer at the idle state is

startup state in the following. Let us define

U

r ≡ remaining startup time for the server begin startup.

Following Borthakur et al. [8], the cumulative distribution function (c.d.f.) of is given by

where is the c.d.f of startup time. Thus, we get the probability density function of remaining startup time for the server startup as

U( ) waits customers in front of him to be served. The expected waiting time of customer at the busy state is

Finally, using the listed above results, we obtain the approximate expected waiting time in the queue given by

1 2

3.5 Comparative Analysis

The primary objective of this section is to examine the accuracy of the maximum entropy results. We present specific numerical comparisons between the exact results and the maximum entropy (approximate) results for the policy M/G/1 queue with general service times, general repair times and general startup times. Conveniently, we represent this queueing system as the policy M/G(G,G)/l queue where the second, third, fourth symbols denote the general distribution of service time, repair time, and startup time, respectively.

N

This section includes the following three subsections:

(i) Comparative analysis for the

N

policy M/M(M,M)/1 and M/D(D,D)/1 queues.

(ii) Comparative analysis for the

N

policy M/E₃(E₄,E₃)/1 and M/M(E₃,E₂)/1 queues.

(iii) Comparative analysis for the

N

policy M/E₃(E₄,D)/1 and M/E₃(E₄,M)/1 queues.

3.5.1 Comparative analysis for the N policy M/M(M,M)/1 and M/D(D,D)/1 queues

Here we perform a comparative analysis between the exact and the approximate (maximum entropy) for the policy M/M(M,M)/1 and M/D(D, D)/1 queues. For the policy M/M(M,M)/1 queue, we obtain

W

q policy M/D(D,D)/1 queue, we have

N

The numerical results are obtained by considering the following parameters:

Case 1: We fix

μ

=1.0,

α

=0.05,

β

=3.0,

γ

=3.0, and vary the values of

λ

from

Case 5: We fixλ =0.3,

μ

=1.0, α =0.05,

β

=3.0, and vary the values of

γ

from 2.0 to 5.0.

Numerical results for the policy M/M(M,M)/1 and M/D(D,D)/1 queues are shown in Table 3.1 for the above five cases. The relative error percentages are very small (0-6.8%).

N

3.5.2 Comparative analysis for the N policy M/E

₃

(E

₄

,E

₃

)/1 and M/M(E

₃

,E

₂

)/1 queues

Here we perform a comparative analysis between the exact and the approximate (maximum entropy) for the policy M/E

W

q queues are shown in Table 3.2 for the above five cases. The relative error percentages are also very small (0-3.5%).

3.5.3 Comparative analysis for the N policy M/ E

₃

(E

₄

,D)/1 and M/E

₃

(E

₄

,M)/1 queues

Here we perform a comparative analysis between the exact and the approximate (maximum entropy) for the policy M/E

W

q shown in Table 3.3 for the above five cases. Again, the relative error percentages are very small (0-3.5%).

Table 3.1. Comparison of exact and approximate for the policy M/M(M,M)/1 and M/D(D,D)/1 queues.

W

_q^*

N

M/M(M,M)/1 M/D(D,D)/1

5 N = N = 10 N = 5 N = 10

Wq W_q^* %Error Wq W_q^* %Error Wq W_q^* %Error Wq W_q^* %Error λ Case 1. (μ=¹^.⁰^, α=⁰^.⁰⁵^, β=³^.⁰^, γ =³^.⁰)

0.2 10.4626 10.4244 0.3657 22.9452 22.8653 0.3481 10.3300 10.3955 0.6345 22.8137 22.8376 0.1049 0.4 5.9040 5.8579 0.7815 12.1359 12.0482 0.7225 5.5494 5.7124 2.9363 11.7834 11.9048 1.0302 0.6 5.1372 5.0756 1.1979 9.2850 9.1820 1.1095 4.3314 4.5880 5.9242 8.4824 8.6975 2.5358 0.8 7.1603 7.0513 1.5226 10.2658 10.1154 1.4654 4.9251 5.2593 6.7861 8.0347 8.3274 3.6437

μ _{Case 2.} (λ=0.3, α=0.05, β=3.0, γ =3.0)

0.5 10.0582 9.9373 1.2022 18.3737 18.1696 1.1107 8.4605 8.9757 6.0898 16.7776 17.2097 2.5752 1.0 7.3178 7.2762 0.5695 15.6334 15.5501 0.5325 7.0903 7.2048 1.6150 15.4074 15.4804 0.4733 1.5 7.0437 7.0179 0.3654 15.3592 15.3057 0.3480 6.9532 6.9967 0.6252 15.2704 15.2861 0.1031 2.0 6.9617 6.9431 0.2683 15.2773 15.2378 0.2583 6.9122 6.9324 0.2924 15.2294 15.2288 0.0038

α Case 3. (λ=0.3, μ=1.0, β=3.0, γ =3.0)

0.05 7.3178 7.2762 0.5695 15.6334 15.5501 0.5325 7.0903 7.2048 1.6150 15.4074 15.4804 0.4733 0.10 7.3384 7.2546 1.1408 15.6539 15.4870 1.0660 7.1006 7.1794 1.1102 15.4177 15.4134 0.0282 0.15 7.3594 7.2333 1.7140 15.6750 15.4241 1.6005 7.1111 7.1540 0.6041 15.4282 15.3464 0.5302 0.20 7.3810 7.2121 2.2890 15.6966 15.3613 2.1359 7.1219 7.1288 0.0968 15.4391 15.2796 1.0328

β Case 4. (λ=0.3, μ=1.0, α=0.05, γ =3.0)

2.0 7.3298 7.2672 0.8549 15.6454 15.5204 0.7991 7.0963 7.1930 1.3625 15.4134 15.4478 0.2226 3.0 7.3178 7.2762 0.5695 15.6334 15.5501 0.5325 7.0903 7.2048 1.6150 15.4074 15.4804 0.4733 4.0 7.3123 7.2811 0.4269 15.6279 15.5655 0.3993 7.0875 7.2109 1.7411 15.4047 15.4969 0.5986 6.0 7.3072 7.2864 0.2845 15.6227 15.5811 0.2661 7.0850 7.2172 1.8670 15.4021 15.5136 0.7238

γ _{Case 5.} (λ=⁰^.³^, μ=¹^.⁰^, α=⁰^.⁰⁵^, β=³^.⁰)

2.0 7.4211 7.3789 0.5685 15.7269 15.6432 0.5323 7.1895 7.3035 1.5858 15.4989 15.5714 0.4675 3.0 7.3178 7.2762 0.5695 15.6334 15.5501 0.5325 7.0903 7.2048 1.6150 15.4074 15.4804 0.4733 4.0 7.2667 7.2253 0.5700 15.5869 15.5039 0.5326 7.0406 7.1553 1.6299 15.3617 15.4348 0.4762 5.0 7.2362 7.1949 0.5702 15.5591 15.4762 0.5327 7.0107 7.1256 1.6390 15.3342 15.4075 0.4779

Table 3.2. Comparison of exact and approximate for the policy M/E

W

_q^*

N

3(E₄,E₃)/1 and M/M(E₃,E₂)/1 queues.

M/E₃(E₄,E₃)/1 M/M(E₃,E₂)/1

5 N = N = 10 N = 5 N = 10

Wq W_q^* %Error Wq W_q^* %Error Wq W_q^* %Error Wq W_q^* %Error λ Case 1. (μ=¹^.⁰^, α=⁰^.⁰⁵^, β=³^.⁰^, γ =³^.⁰)

0.2 10.3742 10.4051 0.2982 22.8575 22.8468 0.0467 10.4611 10.4228 0.3657 22.9442 22.8643 0.3481 0.4 5.6675 5.7607 1.6454 11.9007 11.9524 0.4344 5.9006 5.8545 0.7815 12.1335 12.0459 0.7225 0.6 4.5996 4.7502 3.2733 8.7496 8.8586 1.2465 5.1311 5.0696 1.1981 9.2805 9.1775 1.1095 0.8 5.6692 5.8557 3.2895 8.7774 8.9224 1.6524 7.1482 7.0393 1.5230 10.2557 10.1054 1.4655

μ _{Case 2.} (λ=0.3, α=0.05, β=3.0, γ =3.0)

0.5 8.9927 9.2959 3.3715 17.3093 17.5293 1.2711 10.0537 9.9328 1.2022 18.3700 18.1660 1.1107 1.0 7.1660 7.2285 0.8714 15.4827 15.5035 0.1348 7.3154 7.2737 0.5695 15.6318 15.5485 0.5325 1.5 6.9833 7.0037 0.2921 15.2999 15.2926 0.0478 7.0416 7.0158 0.3654 15.3579 15.3045 0.3480 2.0 6.9287 6.9359 0.1046 15.2453 15.2318 0.0888 6.9598 6.9411 0.2683 15.2761 15.2367 0.2583

α Case 3. (λ=0.3, μ=1.0, β=3.0, γ =3.0)

0.05 7.1660 7.2285 0.8714 15.4827 15.5035 0.1348 7.3154 7.2737 0.5695 15.6318 15.5485 0.5325 0.10 7.1796 7.2043 0.3433 15.4962 15.4377 0.3776 7.3351 7.2514 1.1409 15.6515 15.4846 1.0660 0.15 7.1936 7.1802 0.1863 15.5102 15.3720 0.8907 7.3554 7.2293 1.7141 15.6717 15.4209 1.6005 0.20 7.2079 7.1562 0.7175 15.5245 15.3064 1.4045 7.3761 7.2073 2.2892 15.6925 15.3573 2.1359

β Case 4. (λ=0.3, μ=1.0, α=0.05, γ =3.0)

2.0 7.1739 7.2175 0.6073 15.4905 15.4717 0.1214 7.3264 7.2638 0.8549 15.6427 15.5177 0.7991 3.0 7.1660 7.2285 0.8714 15.4827 15.5035 0.1348 7.3154 7.2737 0.5695 15.6318 15.5485 0.5325 4.0 7.1624 7.2343 1.0033 15.4790 15.5197 0.2628 7.3103 7.2791 0.4269 15.6266 15.5642 0.3993 6.0 7.1590 7.2403 1.1350 15.4756 15.5361 0.3907 7.3054 7.2846 0.2845 15.6217 15.5801 0.2661

γ _{Case 5.} (λ=⁰^.³^, μ=¹^.⁰^, α=⁰^.⁰⁵^, β=³^.⁰)

2.0 7.2666 7.3285 0.8525 15.5748 15.5952 0.1310 7.4166 7.3744 0.5685 15.7242 15.6405 0.5323 3.0 7.1660 7.2285 0.8714 15.4827 15.5035 0.1348 7.3154 7.2737 0.5695 15.6318 15.5485 0.5325 4.0 7.1159 7.1786 0.8811 15.4367 15.4578 0.1367 7.2650 7.2236 0.5700 15.5856 15.5026 0.5326 5.0 7.0858 7.1486 0.8870 15.4091 15.4303 0.1378 7.2348 7.1935 0.5703 15.5580 15.4751 0.5327

Table 3.3. Comparison of exact and approximate for the policy M/E

W

_q^*

N

3(E₄,D)/1 and M/E₃(E₄,M)/1 queues.

M/E₃(E₄,D)/1 M/E₃(E₄,M)/1

5 N = N = 10 N = 5 N = 10

Wq W_q^* %Error Wq W_q^* %Error Wq W_q^* %Error Wq W_q^* %Error λ Case 1. (μ=¹^.⁰^, α=⁰^.⁰⁵^, β=³^.⁰^, γ =³^.⁰)

0.2 10.3734 10.4044 0.2983 22.8571 22.8464 0.0467 10.3756 10.4065 0.2981 22.8582 22.8475 0.0467 0.4 5.6660 5.7593 1.6460 11.9000 11.9517 0.4345 5.6703 5.7636 1.6443 11.9022 11.9539 0.4343 0.6 4.5975 4.7481 3.2753 8.7485 8.8575 1.2467 4.6039 4.7544 3.2693 8.7517 8.8608 1.2459 0.8 5.6664 5.8529 3.2917 8.7760 8.9210 1.6529 5.6748 5.8612 3.2849 8.7803 8.9253 1.6515

μ _{Case 2.} (λ=0.3, α=0.05, β=3.0, γ =3.0)

0.5 8.9916 9.2948 3.3720 17.3087 17.5288 1.2712 8.9949 9.2980 3.3704 17.3104 17.5304 1.2710 1.0 7.1650 7.2274 0.8716 15.4821 15.5030 0.1348 7.1682 7.2307 0.8710 15.4838 15.5046 0.1347 1.5 6.9822 7.0026 0.2922 15.2994 15.2920 0.0478 6.9855 7.0059 0.2919 15.3010 15.2937 0.0479 2.0 6.9276 6.9349 0.1047 15.2448 15.2312 0.0888 6.9309 6.9381 0.1045 15.2464 15.2329 0.0889

α Case 3. (λ=0.3, μ=1.0, β=3.0, γ =3.0)

0.05 7.1650 7.2274 0.8716 15.4821 15.5030 0.1348 7.1682 7.2307 0.8710 15.4838 15.5046 0.1347 0.10 7.1785 7.2032 0.3435 15.4957 15.4372 0.3776 7.1818 7.2064 0.3429 15.4973 15.4388 0.3777 0.15 7.1925 7.1791 0.1861 15.5096 15.3715 0.8907 7.1957 7.1823 0.1867 15.5113 15.3731 0.8908 0.20 7.2068 7.1551 0.7173 15.5239 15.3059 1.4045 7.2100 7.1583 0.7178 15.5256 15.3075 1.4046

β Case 4. (λ=0.3, μ=1.0, α=0.05, γ =3.0)

2.0 7.1728 7.2164 0.6076 15.4900 15.4712 0.1214 7.1761 7.2196 0.6069 15.4916 15.4728 0.1214 3.0 7.1650 7.2274 0.8716 15.4821 15.5030 0.1348 7.1682 7.2307 0.8710 15.4838 15.5046 0.1347 4.0 7.1613 7.2332 1.0035 15.4785 15.5192 0.2628 7.1646 7.2365 1.0029 15.4801 15.5208 0.2627 6.0 7.1579 7.2392 1.1352 15.4751 15.5355 0.3907 7.1612 7.2424 1.1346 15.4767 15.5372 0.3907

γ _{Case 5.} (λ=⁰^.³^, μ=¹^.⁰^, α=⁰^.⁰⁵^, β=³^.⁰)

2.0 7.2642 7.3261 0.8529 15.5736 15.5940 0.1311 7.2714 7.3334 0.8516 15.5773 15.5977 0.1309 3.0 7.1650 7.2274 0.8716 15.4821 15.5030 0.1348 7.1682 7.2307 0.8710 15.4838 15.5046 0.1347 4.0 7.1152 7.1779 0.8812 15.4363 15.4574 0.1367 7.1171 7.1798 0.8809 15.4373 15.4584 0.1366 5.0 7.0854 7.1482 0.8871 15.4089 15.4301 0.1378 7.0866 7.1494 0.8868 15.4095 15.4307 0.1378

Chapter 4 Optimization of the T Policy M/G/1 Queue with Server Breakdowns and Startup Times

We consider a policy M/G/1 queue in which the server is typically subject to unpredictable breakdowns. It is assumed that arriving customers follow a Poisson process and the breakdown times of the server follow the negative exponential distribution. We also assume that the service times, the repair times, and the startup times obey a general distribution. After a period of length T , the server is immediately turned on but is temporarily unavailable to serve waiting customers if there is at least one customer in the waiting line; otherwise, the server waits another period of length and so on until at least one customer is present. When the server turns on, he requires for the preparatory work (i.e. begin startup) before starting service. Once the startup is terminated, the server immediately starts serving the waiting customers. We develop the probability generating function and various system performance measures such as the expected number of customers in the system, the expected length of the idle, busy, and breakdown period, and the expected length of the busy cycle, etc. Based on the derived results, we construct the total expected cost function per unit time, including customer holding cost, the system setup cost, server on and off costs, server startup cost, and server breakdown cost. We determine the optimal threshold numerically to minimize the total expected cost. In addition, numerical results and sensitivity investigations are also presented

T

4.1 Assumptions and Notations

It is assumed that customers arrive according to a Poisson process with parameter λ . The service times of the customers are independent and identically distributed (i.i.d.) random variables obeying an arbitrary distribution function

( ) with a finite mean

S( )

F t t ≥ 0 μ

_S and a finite variance σ_S² . The server is subject to breakdowns at any time with Poisson breakdown rate

α

when he is working. When the server fails, he is immediately repaired at a repair facility, where the repair times are i.i.d. random variables having a general distribution function ( ) with a finite mean

R( )

F t t ≥ 0

μ

R and a finite variance σ . Arriving customers form a single waiting _R² line at a server based on the order of their arrivals. The server can serve only one

customer at a time and the service is independent of the arrival process. 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 the server, customers continue to arrive following a Poisson process. Furthermore, if there is at least one customer in waiting line after a period of length T , the server is immediately turned on (i.e. begin startup) but is temporarily unavailable to serve the waiting customers. He needs a startup time with random length before starting service.

Again, the startup times are i.i.d. random variables obeying a general distribution function

F t

_U( ) (

t ≥ 0

) with a finite mean

μ

_U and a finite variance σ . Once the _U² startup is terminated, the server begins serving the waiting customers until the system becomes empty. Service is allowed to be interrupted if the server breaks down, and the server is immediately repaired. Once the server is repaired, he immediately returns to serve customers until there are no customers in the system.

The following notations and probabilities are used throughout this chapter.

T

− threshold

S −

service time random variable

U −

startup time random variable

R

− repair time random variable

S( )

F

⋅ − distribution function of

S

U( )

F

⋅ − distribution function of

U

R( )

F

⋅ − distribution function of R

A t

( )− number of customers arriving into the system during [0, ]

t A

m− arrival time of the m-th customer

( )

F

⋅ −

distribution function of

A

( )

G z

− p.g.f. of the number of customers waiting in the queue during an idle period

( )

G z p.g.f. of the number of customers arriving during a startup period

( )

f ⋅ −

Laplace-Stieltjes transform (LST) of startup time

W z

( )− p.g.f. of the number of customers that arrive during the turned-off plus the startup period

G z

( )− p.g.f. of the number of customers in the ordinary M/G/1 queue with unreliable server

T( )

G z

− p.g.f. of the number of customers in the policy M/G/1 queue with server breakdowns and general startup times

T

L

T − expected number of customers in the policy M/G/1 queue with server breakdowns and general startup times

T

H

O− complete period of the ordinary M/G/1 queue with server breakdowns

I

T − turned-off period of the T policy M/G/1 queue with server breakdowns and general startup times

U

T − startup period of the T policy M/G/1 queue with server breakdowns and general startup times

B

T − busy period of the policy M/G/1 queue with server breakdowns and general startup times

T

D

T − breakdown period of the T policy M/G/1 queue with server breakdowns and general startup times

H

T − complete period which is equal to (

B

_T +

D

_T)

V

T − complete startup period which is equal to (

U

_T +

H

_T)

C

T − busy cycle which is equal to (

I

_T +

V

_T)

( )

F

⋅ −

distribution function of

V

O( )

F

H ⋅ − distribution function of

H

(_O^{m n})

( )

F

H ⁺

⋅ −

(m n+ ) -fold convolution of

F

_H_O( )⋅

( )

f

⋅ −

LST of

V

P −

probability that the server is idle in the policy M/G/1 queue with server breakdowns and general startup times

T

P −

probability that the server is startup in the policy M/G/1 queue with server breakdowns and general startup times

T

P −

probability that the server is busy in the policy M/G/1 queue with server breakdowns and general startup times

T

P −

probability that the server is broken down in the policy M/G/1 queue with server breakdowns and general startup times

T

C

h− holding cost per unit time for each customer present in the system;

C

s− setup cost for per busy cycle;

C

i− cost per unit time for keeping the server off;

C

−

startup cost per unit time for the preparatory work of the server before starting the service;

C

b− cost per unit time for keeping the server on and in operation;

C

d − breakdown cost per unit time for a failed server

4.2 System Performance Measures

In this section, we focus mainly on developing some important system performance measures, such as (i) the expected number of customers in the system; (ii) the expected length of the idle period, the complete startup period, the busy period, and the breakdown period; (iii) the expected length of the busy cycle; and (iv) the probability that the server is idle, startup, busy and broken down.

4.2.1 Expected number of customers in the system

Let be a random variable representing the completion time of a customer, which includes both the service time of a customer and the repair time of a server.

Applying the well-known results of Medhi and Templeton [34], the probability

在文檔中在N方策和T方策下具故障及啟動服務者M/G/1排隊之研究 (頁 32-0)