Scope of Dissertation

The objective of this dissertation is fourfold: First, we analyze an unreliable server in the and

T

policies M/G/1 queue with general repair times and startup time. We derive the steady state queue length distribution and obtain various system performance measures. Second, optimal threshold and to minimize the overall operating cost of the system are found, respectively. Third, we present sensitivity analysis and some numerical computations to verify the analytical results and show how to make the decision based on minimizing the cost function. Finally, we apply the maximum entropy principle to develop the maximum entropy solutions

N

N T

and perform a comparative analysis between the maximum entropy results and exact results.

Chapter 1 is an introduction. In Chapter 1, we review some controllable queues and some techniques relevant to this study such as M/G/l decomposition property and maximum entropy principle.

In Chapter 2, we study the policy M/G/1 queue with server breakdowns and general startup times. We first develop various system performance measures, such as the expected number of customers, the expected length of the turned-off, complete startup, busy, and breakdown periods. Next, we construct the total expected cost function per unit time to determine the optimal threshold numerically in order to minimize the cost function. In addition, the analytic results for sensitivity analysis are derived. Furthermore, we investigate some numerical examples.

N

N

In Chapter 3, we use maximum entropy principle to 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. The purpose of this chapter is: (i) to provide the maximum entropy formalism for the policy M/G/1 queue with general repair times and general startup times; (ii) to develop the maximum entropy (approximate) solutions for the policy M/G/1 queue with general repair times and general startup times by using Lagrange’s method;

(iii) to obtain approximate results for the expected waiting time in queue; (iv) to perform a comparative analysis between the approximate results with established exact results for various distributions, such as exponential (M), k-stage Erlang (E

N

N

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 system.

In Chapter 4, 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, turn the 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

In Chapter 5, we use maximum entropy principle to study a single removable and unreliable server in the policy M/G/1 queue with general repair times and general startup times. First, we develop the maximum entropy (approximate) solutions for the policy M/G/1 queue with general repair times and general startup times by using Lagrange’s method under the constraint “the first moment of the queue length and then to obtain approximate results for the expected waiting time in queue. Next, we develop the maximum entropy solutions under the constraint “the second moment of the queue length” and then to obtain approximate results for the expected waiting time in queue. Finally, we perform a comparative analysis between the approximate results with established exact results for various distributions under the constraints of first moment and second moment of the queue length.

T

T

Chapter 6 presents some conclusions based on results of the study, and recommendations for further investigations.

Chapter 2

Optimization of the N Policy M/G/1 Queue with Server Breakdowns and Startup Times

In this chapter, we deal with the policy M/G/1 queue with a single removable and unreliable server whose arrivals form a Poisson process. Service times, repair times, and startup times are assumed to be generally distributed. When the queue length reaches

(

), the server is immediately turned on but is temporarily unavailable to serve the waiting customers. The server needs a startup time before providing service until there are no customers in the system. Firstly, we develop various system performance measures, such as the expected number of customers, the expected length of the turned-off, complete startup, busy, and breakdown periods. Next, we construct the total expected cost function per unit time to determine the optimal threshold numerically in order to minimize the cost function In addition, sensitivity analysis and some numerical examples are also investigated

N

N N

≥1

N

2.1 Assumptions and Notations

It is assumed that arrivals of customers follow a Poisson process with rate

λ

. The service times for a customer are independent and identically distributed (i.i.d.) random variables obeying an arbitrary distribution function ( ) with a mean service time

S( )

F t t

≥0

μ

S and a finite variance σ . The server is subject to breakdowns _S² 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 mean repair time

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, when the queue length reaches a specific level, denoted by , the server is immediately turned on (i.e. begin startup) but is temporarily unavailable to serve the waiting

N

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

( ), with a mean startup time

U( )

F t t

≥0

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.

In this chapter, the following notations and probabilities are used.

N

− 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 ( )

G z −

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

N( )

G z

− p.g.f. of the number of customers in the policy M/G/1 queue 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

H

O− complete period (busy period plus breakdown period) of the ordinary M/G/1 queue with server breakdowns

I

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

N

U

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

N

B

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

N

D

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

N

H

N − complete period which is equal to (

B

_N +

D

_N)

V

N− complete startup period which is equal to (

U

_N +

H

_N)

C

N − busy cycle which is equal to (

I

_N +

V

_N)

N( )

F

V ⋅ − distribution function of

V

_N

O( )

F

H ⋅ − distribution function of

H

_O

(_O^{N n})( )

F

H ⁺ ⋅ −

(N n + )

-fold convolution of

F

H_O( )⋅

N( )

f

V ⋅ − LST of

V

_N

IN

P −

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

N

UN

P −

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

N

BN

P

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

N

DN

P

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

N

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

sp

−

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

2.2 Justification of Practical Applications

A number of practical problems arise which may be formulated as one in which the server meets unpredictable breakdowns and requires a startup time before providing service. Such models have potentially useful in practical production (manufacturing) systems. For example, in reflow work for Printed Circuit Board (PCB) Surface Mount. Assume that PCB arrives according to a random process. For cost concern, it is desirable that the reflow machine begins operating whenever the number of PCB reaches a critical value . It takes random time for warming up before the reflow machine starts working. Moreover, the reflow process may be interrupted when machine encounters unpredicted breakdowns. When reflow interruptions occur (breakdowns), it is emergently recovered with a random time.

Another possible application is wire bonding in Integrated Circuit (IC) assembly. To save cost, it is desirable that the wire bonder begins operating whenever the number of unbounded IC reaches a critical value . It requires a random time for setup before the wire bonder starts working. The bonding process may be interrupted when the bonder meets breakdowns. When bonding interruptions occur (breakdowns), it is emergently recovered.

N

N

2.3 System Performance Measures

The primary objective of this section is to develop the various system performance measures, such as (i) expected number of customers in the system; (ii) expected length of the turned-off period, the complete startup period, the busy period, and the breakdown period; (iii) expected length of the busy cycle; and (iv) the probability that the server is turned-of, startup, busy and broken down

2.3.1 Expected number of customers in the system

Let

H 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 ordinary M/G/1 queue with unreliable server is given by

(1 )(1 ) ( )

( ) ( )

H H

H

z f z

G z f z z

ρ λ λ

λ λ

− − −

= − − , (2.1)

where [ ]

ρ

_H =

λ E H

. In addition, [ ]

E H

=

μ

_S(1+

αμ

_R) be less than unity. We note that expression (2.1) is obtained only by replacing service times by completion times in the formula of the ordinary M/G/1 queue with reliable server.

For the policy M/G/1 queue with server breakdowns requiring startup time, we consider that the server is on 'extended vacation' during the turned-off period plus the startup period. Following the result of Medhi and Templeton [34], we obtain

N

server breakdowns and general startup times;

N

( )

W z ≡

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

≡

[the p.g.f. of the number of customers that arrive during the turned-off period]

×

[the p.g.f. of the number of customers that arrive during the startup period];

Let denote the expected number of customers in the policy M/G/1 queue with server breakdowns and general startup times. Thus we have

L

N

N

( ) 1

2.3.2 Expected length of the turned-off, complete startup, busy, and breakdown periods

The turned-off period terminates when the -th customer arrives in system.

Since the complete startup period starts when the turned-off period terminates, the complete startup period is represented by the sum of the startup period and the complete period. The server begins startup when there are at least

waiting

customers in the system. This is called the startup period. The startup period terminates when the server starts to serve the waiting customers. Since the complete period begins when the startup period is over and terminates when the system becomes empty, the complete period is represented by the sum of the busy period and the breakdown period. The busy period is initiated when the server completes his startup and begins serving the waiting customers. During the busy period, the server may break down and starts his repair immediately. This is call the breakdown period.

After the server is repaired, he returns immediately and provides service until there are no customers in the system.

N

N

Let be the complete period of the ordinary M/G/1 queue with server breakdowns. Using the well-known result of Kleinrock [28, p. 213], we obtain the expected length of the complete period for the ordinary M/G/1 queue with server breakdowns as

2.3.2.1 Expected length of the turned-off period

We know that the turned-off period

I terminates when the

_N -th customer arrives in system. Since the length of times between two successive arrivals are independently, identically and exponentially distributed with mean

N

1

λ

, thus the expected length of the turned-off period, , for the policy M/G/1 queue with server breakdowns

[ ]_N

E I N

[ ]

_N

N

E I ⁼

λ . (2.5)

2.3.2.2 Expected length of the complete startup period

Let represent the complete startup period for the policy M/G/1 queue with server breakdowns and general startup times. Since the complete startup period is the sum of the complete period and the startup period which implies

, where and denote the complete period and the startup

f

V

⋅

be the LST of the distribution of the complete startup period of the

N

policy M/G/1 queue with server breakdowns.

The following notations are used.

N

( )

F

V

⋅ −

distribution function of the complete startup period of the policy M/G/1 queue with server breakdowns and general startup times.

V

N

N

U

( )

f ⋅ −

the LST of startup time

O( )

F

H ⋅ − distribution function of the complete period of the ordinary M/G/1 queue with server breakdowns.

H

O

F x P given any startup time t completion startup period generated by N customers arrival plus n customers arrival in the complete period H during t x t dF t

∞

Taking the LST of both sides of (2.6) yields

( )

Changing the order of integration of (2.7), it finally gets

0

Differentiating (2.8) with respect to , we obtain the expected length of the complete startup period as follows:

s

2.3.2.3 Expected length of the busy and breakdown periods

The expected length of the complete period and the expected length of the startup period are denoted by and , respectively. Recall that

which implies expected length of the breakdown period, respectively. Recall that the complete period is the sum of the busy period and the breakdown period which implies

= + . Hence from (2.10) we have

2.3.3 Expected length of the busy cycle

The busy cycle for the

policy M/G/1 queue with server breakdowns and

general startup times, denoted by , is the length of time from the beginning of the last turned-off period to the beginning of the next turned-off period. Since the busy cycle is the sum of the turned-off period (

N

2.3.4 Probability that the server is turned-off, startup, busy and broken down

　In steady-state, let

IN

P ≡

probability that the server is turned-off.

UN

P ≡

probability that the server is startup.

N ≡

P

B probability that the server is busy.

N ≡

P

D probability that the server is broken down.

We obtain (2.13), and in (2.15) into relations (2.16)-(2.19) yields the probability that the server is turned-off, startup, busy and broken down in the following:　

[ ]_N

(1 )

We prove from (2.22) that the probability that the server is busy in the steady-state is equal toρ .

2.4 The Optimal N Policy

We develop an expected cost function per unit time for the

policy M/G/1

queue with server startup and breakdowns in which

is a decision variable. Our

objective is to determine the optimum value of the control parameter , say , so as to minimize this function. We define the following cost elements:

N N

N N

^*

C

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

s ≡

C

setup cost for per busy cycle;

i ≡

C

cost per unit time for keeping the server off;

C

sp

≡

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

b ≡

C

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

d ≡

C

breakdown cost per unit time for a failed server.

Utilizing the definition of each cost element listed above, the expected cost function per unit time per customer is given by

[ ] [ ] [ ] [ ]

E C do not involve the decision variable . Omitting these cost terms are not functions of the decision variable . The optimization problem in (2.24) is equivalent to minimize the following equation:

N

N

( )

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

3

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

,

sp

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

,

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,

_s

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

h

C =1000,

s

C

_h=5 and change the specified values of (

C

_sp, ).

Table 2.5 reveals that is insensitive to (

C

i

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

h

C

_b

C

_i

C

_d

C =200. In addition, we fix startup cost

_s

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,

s

C

_h=5,

C

_sp= 100,

C

_i=60

N

*

λ

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,

s

C

_h=5,

C

_sp= 100,

C

_i=60

N

*

μ

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,

_s

h=5,

C C

_sp= 100,

C

_i=60

N

*

α

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,

s

C

_h=5,

C

_sp= 100,

C

_i=60

N

*

β

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,

_s

C

_h=5,

C

sp= 100,

C

_i=60

N

*

γ

Figure 2.6 Plots of (

γ

, N^*) with

λ

=0.3, μ =1,

α

=0.05, β =3,

C =500,

_s

C

_h=5,

C

sp= 100,

C

_i=40

N

*

γ

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

_O

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,

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,

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