Scope of Dissertation

In chapter 2, we provide a recursive method using the supplementary variable technique to derive the steady-state probability distributions in the F policy M/G/1/K queue. We illustrate the solution algorithm by presenting three simple examples for three different service time distributions: exponential (denoted M), 3-stage Erlang (denoted E3), and deterministic (denoted D). Various system performance measures are also presented. The total expected cost function per unit time is developed. Numerical and comparative results are shown. In chapter 3, we follow the above method of the chapter 2 and treating the supplementary variable as the remaining interarrival time to develop the F policy G/M/1/K queue with startup time. In chapter 4, we study N policy M/G/1/K queue with startup time. In chapter 5, we examine the relationships between the F policy and N policy. Using the solution algorithm of N policy (F policy) M/G/1/K queue with startup time, we develop the steady state probabilities of F policy (N policy) G/M/1/K queue.

Finally, chapter 6 consists of some concluding remarks.

Chapter 2

The F Policy M/G/1/K Queue with Startup Time

A supplementary variable technique is used to study the optimal management problem of the F policy M/G/1/K queue where the server needs a startup time before start allowing customers in the system and K < ∞ denotes the maximum number of customers in the system. The method of controlling arrivals focuses on reducing the number of customers in the system. The model presented in this dissertation is very useful in real-life situations since the controlling of arriving customers is considered.

The primary objective of this chapter is threefold. Firstly, we develop a recursive method using the supplementary variable technique and treating the supplementary variable as the remaining service time, to develop the steady-state probability distributions of the number of customers for the F policy M/G/1/K queue. The method can be utilized for any service time distribution, such as deterministic (denoted D), exponential (denoted M) and k-stage Erlang (denoted Ek), etc. Secondly, to illustrate a recursive method we present three simple examples for three different service time distributions such as exponential, 3-stage Erlang, and deterministic. Thirdly, we study various system performance measures, such as the average number of customers in the system, the probability that the server is busy, the blocking probability, etc. The total expected cost function per unit time for the F policy M/G/1/K queue with startup times is developed. Numerical and comparative results are also provided.

2.1 Assumptions and Notations

We consider the controlling arrivals to a finite capacity M/G/1 queue with combined F policy and exponential startup time. It is assumed that customers arrive according to a Poisson process with parameter λ , and the service times of the successive customers are independently and identically distributed (i.i.d.) random variables having a distribution

S u ( ) ( u ≥

⁰

)

, a probability density function

s u ( )

( u ≥

⁰

)

and mean service time s . The arrival process is independent of the service ₁ process. We assume that arriving customers form a single waiting line based on the order of their arrivals; that is, the first-come, first-served discipline. Suppose that the

server can serve only one customer at a time. Customers entering into the service facility and finding that the server is busy have to wait in the queue until the server is available. Gupta [6] first introduced the concept of a F policy. The definition of a F policy is described as follows: When the number of customers in the system reaches its capacity K (i.e. the system becomes full), no further arriving customers are allowed to enter the system until there are enough customers in the system have been served so that the number of customers in the system decreases to a threshold value F (0≤ <

F K

−1). At that time, the server requires to take an exponential startup time with parameter

β

to start allowing customers in the system. Thus, the system operates normally until the number of customers in the system reaches its capacity at which time the above process is repeated all over again.

The following notations and probabilities are used throughout this chapter.

F threshold level

K system capacity

( K

> +

F

¹

)

S

service time random variable

U

remaining service time random variable

( )

S u

distribution function (d.f.) of

S

( )

s u

probability density function (p.d.f.) of

S

( )

S*

θ

Laplace-Stieltjes transform (LST) of

S ( ) ( )

S

* l

θ l

th order derivative of ^S*

( ) ^θ

with respect to

θ

0,0

( )

P t

probability of no customers in the system at time t when the arrivals are not allowed to enter the system

0,n

( )

P t

probability of

n

customers in the system at time t when the arrivals are not allowed to enter the system, where n=1,2,..., .K

( )

P

1,0

t

probability of no customers in the system at time t when the arrivals are allowed to enter the system

1,n

( )

P t

probability of

n

customers in the system at time t when the arrivals are allowed to enter the system, where n=1,2,...,K − 1.

P 0,0 steady state probability of no customers in the system when the arrivals are not allowed to enter the system

P 0,n steady state probability of

n

customers in the system when the arrivals are not allowed to enter the system, where n=1,2,..., .K P 1,0 steady state probability of no customers in the system when the

arrivals are allowed to enter the system

P 1,n steady state probability of

n

customers in the system when the arrivals are allowed to enter the system, where n=1,2,...,K − 1.

s 1 mean service time

The special case with system capacity K=F+1 is presented in the appendix.

2.2 Development of the Equations and Solutions

We use the following supplementary variable:

U ≡

remaining service time for the customer in service. The state of the system at time t is given by

( )

N t ≡

number of customers in the system, and

( )

U t ≡

remaining service time for the customer being served.

Let us define

2.3 Steady State Results

In steady state, let us define

→∞

( )

From (2.1)-(2.10), we can easily obtain the following steady state equations:

0,0 0,1

( )

and

Therefore, we take the LST on both sides of (2.12)-(2.14) and (2.16)-(2.19). It yields

( ) ( ) ( ) ( ) ( )

where

Solving (2.36)-(2.38) recursively, we obtain

( )

^λ

( )

^{λ ζ β − +}_λ^φ

( )

^{λ − +}_λ

( )

^λ +

( )

Using (2.39) in (2.34), we can obtain

( ) ( ) ( ) ( )

Remark: The representative meaning of the above formulation (2.42) is to sum up all possible products of k κ in which the total of subscript values of s

κ

equals n.

We give an easily understood example for n = 4:

4 4 3 1 2 2 1 3 1 1 2 1 2 1 2 1 1 1 1 1 1

Using (2.42) and (2.43) to solve (2.41) recursively, and including (2.15) and (2.33), we finally get

( ) ( )

Substituting (2.45), (2.44), and (2.35) into (2.39) finally yields

( ) ( ) ( ) ( )

As

P

1,1^{* 1}

^{( )} ( )

0 is known completely from (2.52), the values

P

1,^{* 1}_n

^{( )} ( )

0 for

Substituting (2.55) into (2.51), we have

( )

^{1 * 1}

^{( )} ( )

^{* 1}

^{( )} ( )

¹

( )

^{* 1}

^{( )} ( )

determined using the normalizing condition

1

To demonstrate the working of the recursive method, we first describe the solution algorithm for calculating the steady state probabilities, P0,^*_n

( ) (

0 0≤ ≤n K

)

and

( ) ( )

1,*_n 0 0 1

P ≤ ≤n K− . Next, to illustrate the solution algorithm, we provide three simple examples where the service time distributions are exponential, k-stage Erlang, and deterministic, respectively.

2.3.2 The solution algorithm

Let F be the threshold, K be the maximum capacity of the system, and let

( ) ( )

S

* l

θ

be the l-th derivative of ^S*

( ) ^θ

^{, where l =1, 2,...,K}. We set the values of F, K, and the LST expression of the service time distribution, namely ^S*

( ) ^θ

^{. The}

steps of the solution algorithm are stated as follows:

Step 1. For each n=0, 1,...,K , compute

φ

_n using (2.32).

Step 6. For each n=1, 2,...,K− , compute 1

P

1,_n

( )

0 using (2.44) and (2.45) in

We use the solution algorithm to illustrate a recursive method. We provide three simple examples for three different service time distributions such as exponential, 3-stage Erlang, and deterministic, respectively.

Example 1 (For M/M/1 queue). We set the mean service time s₁=1

μ

, where

μ

From (2.31), we finally get

( )

σ μ λ

= .

For each

n =

1, we find from (2.43) that

^κ

^{1 = + +}

(

¹

^{σ σ}

²

) ^{σ (}

¹⁺

^{σ )}

^.

Step 4. For each n =0, 1, compute Ψ . _n

It implies from (2.42) that Ψ = and ₀ 1 ^{Ψ = + +1}

(

¹

^{σ σ}

²

) ^{σ (}

¹⁺

^{σ )}

^.

Step 5. For each n =1, 2, 3, compute

^{A n} ( )

^and

^{B n} ( )

^.

Using (2.46) and (2.47), it follows that

( )

¹

It yields from (2.44) and (2.45) that

( ) ( ) ( )

From (2.48), we finally have

( ) (

^{2 3}

)

Using (2.49) and (2.50) yields

( ) ( ) (

²

)

and *

( ) ( )

Using (2.56), it follows that

( ) ( )

It is to be noted that these results are the same as those given in Gupta [12, p1006].

Example 2 (For M/E3/1 queue). The 3-stage Erlang distribution is made up of three independent and identical exponential stages, each with mean 1 3

μ

. We set the mean service time s₁ =1

μ

, 1F = , and

K =

3. In this case, we have From (2.32), we finally obtain

0 1

φ

= ,

φ

1=3 1

(

−

γ γ )

, and

φ

2 =

φ

3 =3 1

(

−

γ )(

3 2−

γ γ )

², where

^γ

⁼³

^μ (

³

^{μ β}

⁺

)

^.

Step 2. For each n =1, 2, compute P0,^*_n

( )

0 using (2.31) in terms of P . _0,0 From (2.31), it follows that

( )

Step 4. For each

n =

0, compute Ψ . _n It implies from (2.42) that Ψ = . ₀ 1

Step 5. For each n =1, 2, compute

^{A n} ( )

^and

^{B n} ( )

^.

It yields from (2.46) and (2.47) that

( )

^{1 3}

It implies from (2.48) that

( ) ( )( ) ( )

Using (2.49) and (2.50) yields

( ) ( )( )( ) ( )

Step 9. For

n =

3, compute P0,^*_n

( )

0 using (2.56) in terms of P . _0,0 It follows from (2.56) that

( ) ( )( ) ( )

Using (2.31), we finally get

( )

From (2.46) and (2.47), it follows that

( )

¹

Using (2.44) and (2.45) yields

( ) ( ) ( )

Using (2.49) and (2.50) yields

( ) ( ) ( )

It follows from (2.56) that

( ) ( ) ( ) ( )

( )

Our analysis is based on the following system performance measures of the F policy M/G/1/K queue with exponential startup time. Let

L ≡ the average number of customers in the system; s

P ≡ the probability that the server is busy; b

P ≡ the probability that the server requires a startup time before starting the s

service;

P ≡ the probability that the server is blocked. bl

The expressions for L , _s P , _b P , and _s P are give by _bl

We develop the total expected cost function per unit time for the F policy M/G/1/K queue with startup times, in which F is a management decision variable. The main purpose of this subsection is to determine the optimum management F policy so as to minimize this total expected cost function. Let

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

C ≡ busy cost per unit time for a busy server; b

C ≡ startup cost per unit time for the preparatory work of the server before s

starting the service;

C ≡ fixed cost for every lost customer when the system is blocked. bl

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

( ) _{h s b b s s bl bl}

TC F =C L +C P +C P C+

λ

P . (2.58)

The optimal value of F , F is determined by satisfying the following ^* inequality

* *

( 1) ( )

TC F − ≥TC F and TC F( ^*+ ≥1) TC F( ^*). (2.59)

2.5.2 Numerical examples

We now perform a sensitivity analysis on the optimum value F based on ^* changes in specific values of the system parameters and fix the system capacity K=15. We consider the three simple examples for three different service time distributions such as exponential, 3-stage Erlang, and deterministic and employ the following cost elements:

Case 1: C_h =5, C_b =200, C_s =250, C_bl =300.

Case 2: C_h =5, C_b =200, C_s =250, C_bl =350.

Case 3: C_h =5, C_b =200, C_s =300, C_bl =350.

Case 4: C_h =5, C_b =225, C_s =300, C_bl =350.

Case 5: C_h =10, C_b =225, C_s =300, C_bl =350.

In this section we provide the numerical results of the optimal value F and ^* the minimum expected cost for three interarrival time distributions and specific values of λ ,

μ

,

β

. We first fix

(

μ β = (1.0, 3.0) and choose different values of ^,

)

λ = 0.5, 0.6, 0.7. Next, we fix

(

λ β = (0.8, 3.0) and consider various values of ^,

)

μ

= 1.0, 1.1, 1.2. Finally, we fix

(

λ μ = (0.8, 1.0) and select different values of ^,

)

β

= 2.0, 4.0, 5.0.

The optimal value of F, F , and its minimum expected cost ^{* TC F}

( )

^{* for the}

above five cases are shown in Tables 1-3. For fixed values of

(

μ β and various ^,

)

values of λ in Tables 1-3, we observe that (i) ^{TC F}

( )

^* increases as λ increases for any case; and (ii) F decreases as λ increases for any case. For fixed values ^* of

(

λ β and various values of ^,

) μ

in Tables 1-3, we find that (i) ^{TC F}

( )

^*

decreases as

μ

increases for any case; and (ii) F increases as ^*

μ

increases for any case. Again, for fixed

(

λ μ and various values of ^,

) β

in Tables 1-3, we observe that (i) ^{TC F}

( )

^* slightly decreases as

β

increases for any case; and (ii)

F does not change at all when *

β

changes from 2.0 to 5.0 for any case.

Intuitively, F is insensitive to changes in ^*

β

.

It can be easily see from Tables 1 through 3 that (i) F increases as ^* C _h decreases or C increases (see cases 4-5 and cases 1-2); and (ii) _bl C and _h C _bl have a larger effect on F than ^* C and C (see cases 3-4 and cases 2-3).

Table 1. The optimal value of F and its minimum expected cost for the service time distribution such as exponential.

λ (^{μ β,} ) (^{= 1.0,3.0}) ^μ (^{λ β,} ) (^{= 0.8,3.0}) ^β (^{λ μ,} ) (^{= 0.8,1.0})

0.5 0.6 0.7 1.0 1.1 1.2 2.0 4.0 5.0

Case1 F^* 9 7 5 4 7 10 5 4 4

( *)

TC F 105.000 127.486 151.420 177.597 158.454 143.314 177.680 177.561 177.540

Case2 F^* 12 11 9 6 10 12 6 6 6

( *)

TC F 105.001 127.501 151.554 178.285 158.655 143.367 178.361 178.247 178.225

Case3 F^* 12 11 8 6 10 12 6 6 6

( *)

TC F 105.001 127.502 151.562 178.314 158.669 143.374 178.404 178.269 178.242

Case4 F^* 11 9 7 4 8 11 5 5 5

( *)

TC F 117.500 142.496 169.000 197.985 176.767 160.020 198.072 197.941 197.915

Case5 F^* 5 4 3 2 4 6 2 2 2

( *)

TC F 122.470 149.933 180.049 213.873 189.095 169.773 213.960 213.830 213.804

Table 2. The optimal value of F and its minimum expected cost for the service time distribution such as 3-stage Erlang.

λ (^{μ β,} ) (^{= 1.0,3.0}) ^μ (^{λ β,} ) (^{= 0.8,3.0}) ^β (^{λ μ,} ) (^{= 0.8,1.0})

0.5 0.6 0.7 1.0 1.1 1.2 2.0 4.0 5.0

Case1 F^* 9 7 6 4 7 10 4 4 4

( *)

TC F 104.167 125.999 148.912 173.998 155.504 141.111 174.022 173.986 173.979

Case2 F^* 12 11 9 6 10 12 6 6 6

( *)

TC F 104.167 126.000 148.932 174.216 155.541 141.116 174.241 174.204 174.197

Case3 F^* 12 11 9 6 10 12 6 6 6

( *)

TC F 104.167 126.000 148.933 174.226 155.544 141.117 174.255 174.211 174.202

Case4 F^* 11 9 7 5 9 12 5 5 5

( *)

TC F 116.667 141.000 166.424 194.121 173.710 157.781 194.150 194.107 194.099

Case5 F^* 6 4 3 2 4 6 2 2 2

( *)

TC F 120.833 146.996 175.278 207.572 183.632 165.532 207.601 207.557 207.548

Table 3. The optimal value of F and its minimum expected cost for the service time distribution such as deterministic.

λ (^{μ β,} ) (^{= 1.0,3.0}) ^μ (^{λ β,} ) (^{= 0.8,3.0}) ^β (^{λ μ,} ) (^{= 0.8,1.0})

0.5 0.6 0.7 1.0 1.1 1.2 2.0 4.0 5.0

Case1 F^* 10 8 6 4 7 10 4 4 4

( *)

TC F 103.750 125.250 147.578 171.798 153.930 140.000 171.806 171.794 171.792

Case2 F^* 12 11 9 6 10 12 6 6 6

( *)

TC F 103.750 125.250 147.582 171.869 153.938 140.001 171.877 171.865 171.863

Case3 F^* 12 11 9 6 10 12 6 6 6

( *)

TC F 103.750 125.250 147.582 171.872 153.938 140.001 171.882 171.867 171.864

Case4 F^* 12 10 7 5 9 12 5 5 5

( *)

TC F 116.250 140.250 165.080 191.839 172.117 156.667 191.848 191.834 191.831

Case5 F^* 7 5 3 2 4 6 2 2 2

( *)

TC F 120.000 145.500 172.649 203.459 180.568 163.331 203.469 203.454 203.451

Chapter 3

The F Policy G/M/1/K Queue with Startup Time

We use a supplementary variable technique to analyze the optimal control of the F policy G/M/1/K queue where the server needs a startup time before start allowing customers in the system and K < ∞ denotes the maximum capacity of the system. The method of controlling arrivals focuses on reducing the number of customers in the system. The model proposed in this dissertation is very useful in real-life situations since the controlling of arriving customers is considered.

In section 3.1, the queue model is briefly described. Section 3.2 develops the equations and solutions. Section 3.3 provides a recursive method using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to obtain the steady-state probability distributions of the number of customers in the F policy G/M/1/K queue. In section 3.4, we illustrate the solution algorithm by presenting three simple examples for three different interarrival time distributions: exponential (denoted M), 3-stage Erlang (denoted E3), and deterministic (denoted D). In section 3.5, various system performance measures are presented. The total expected cost function per unit time for the F policy G/M/1/K queue with startup times is developed. Numerical and comparative results are also provided.

3.1 Assumptions and Notations

We consider the category of controlling the arrivals to the F policy G/M/1/K queue with exponential startup time. It is assumed that the times elapsing between successive arrivals are independently and identically distributed (i.i.d) random variables having general distribution

^{A v} ( ) ( ^{v ≥}

⁰

)

, a probability density function

a v ( ) ( v ≥

⁰

)

and mean interarrival time b . The service times of ₁ the customers are independently random variables having a common exponential distribution with mean 1 μ . Let us assume that customers arriving at the server form a single waiting line and are served in the order of their arrivals; that is, according to the first-come, first-served (FCFS) discipline. Suppose that the server can serve only one customer at a time, and that the service is independent of the arrival of the customers. Customers entering into the service facility and finding that

the server is busy have to wait in the queue until the server is available. Gupta [6]

first introduced the concept of a F policy. The definition of a F policy is described as follows: When the number of customers in the system reaches its capacity K (i.e. the system becomes full), no further arriving customers are allowed to enter the system until there are enough customers in the system have been served so that the number of customers in the system decreases to a threshold value F (0≤ <

F K

−1). At that time, the server needs to take an exponential startup time with parameter

β

to start allowing customers in the system. Thus, the system operates normally until the number of customers in the system reaches its capacity at which time the above process is repeated all over again.

The following notations and probabilities are used throughout this chapter.

F threshold level

K system capacity

( ^K

^{> +}

^F

¹

)

A Interarrival time random variable

V remaining interarrival time random variable

( )

A v

distribution function (d.f.) of A

( )

a v

probability density function (p.d.f.) of A

( )

a*

θ

Laplace-Stieltjes transform (LST) of A

( ) ( )

a

* l θ

l

th order derivative of a^*

( ) θ

with respect to θ

0,0

( )

P t

probability of no customers in the system at time t when the arrivals are not allowed to enter the system

0,n

( )

P t

probability of

n

customers in the system at time t when the arrivals are not allowed to enter the system, where n=1,2, ..., .K

1,0

( )

P t

probability of no customers in the system at time t when the arrivals are allowed to enter the system

1,n

( )

P t

probability of

n

customers in the system at time t when the arrivals are allowed to enter the system, where n=1,2, ...,K − 1.

The special case with system capacity K=F+1 is presented in the appendix.

3.2 Development of the Equations and Solutions

We use the following supplementary variable:

V ≡

remaining interarrival time for the customer in arrival process. The state of the system at time t is given by

( )

N t ≡

number of customers in the system, and

( )

V t ≡

remaining interarrival time for the customer who is arriving.

Let us define

( )

⁼

{ ( )

^{= <}

( )

^{≤ +}

}

^{≥ =} In steady state, let us define

→∞

( )

From (3.1)-(3.8) we can easily obtain the following steady state equations:

0,0 0,1

Therefore if the LST is taken on both sides of (3.14)-(3.17), it is found that

( ) ( ) ( ) ( )

where

φ β β

^ζ

Solving (3.27)-(3.28) recursively, we obtain

( ) ( ) ( ) ( )

( ) ^{( )} ( )

Using (3.28)-(3.29) in (3.25), we can obtain

( ) ( )

The representative meaning of the above formulation (3.32) is the same as (2.42).

Using (3.32)-(3.33) to solve (3.31) recursively, and including (3.26), we finally get

( )

^{− −} − − −

( )

( ) _μ

−

( ) ^β

^{ζ −}

^φ

determined recursively using (3.37) in terms of P . _0,0

Now the only unknown quantity is P1,0^*

( )

0 which can be obtained from determined using the normalizing condition

1

To demonstrate the working of the recursive method, we first describe the solution algorithm for calculating the steady state probabilities,

( ) ( )

0,*_n 0 0

P ≤ ≤n K and P1,^*_n

( ) (

0 0≤ ≤n K− . Next, to illustrate the solution 1

)

algorithm, we provide three simple examples where the interarrival time distributions are exponential, k-stage Erlang, and deterministic, respectively.

3.3.2 The solution algorithm

Let F be the threshold, K be the maximum capacity of the system, and let

( ) ( )

a

* l

θ

where l =1, 2,...,K be the lth derivative of a^*

( ) θ

. We are given the values of F, K, and the LST expression of the interarrival time distribution, namely

( )

a*

θ

. The steps of the solution algorithm are stated as follows:

Step 1. For each n=0, 1,...,K , compute

φ

_n using (3.24).

Step 2. For each n=1, 2,...,K , compute P0,^*_n

( )

0 using (3.22) in terms of

We use the solution algorithm to illustrate a recursive method. We provide three simple examples for three different interarrival time distributions such as exponential, 3-stage Erlang, and deterministic, respectively.

Example 1 (For M/M/1 queue). We set

λ

is the interarrival rate. Assume that 2 Using (3.22), we finally get

( )

( ) ( ) ( )

It implies from (3.31) that

0 1

It follows from (3.36) that

( ) ( ) ( )

Step 7. For each n =1, 2, , 4" , compute P1,^*_n

( )

0 using (3.37) in terms of P . _0,0

It is to be noted that these results are the same as those given in Gupta [12, p1006].

Example 2 (For E3/M/1 queue). The 3-stage Erlang distribution is made up of three independent and identical exponential stages, each with mean 1 3

λ

. We set

λ

is the interarrival rate, F = , and 1

K =

3. In this case, we have Using (3.22), we finally get

( )

Step 3. Compute A₁ and

κ

₁using (3.30) and (3.33), respectively.

It follows from (3.36) that

( ) ( )

²

Step 9. Determine P using (3.41). Thus _0,0 P0,^*_n

( ) (

0 n =1, 2, 3

)

are achieved from Step Using (3.22), we finally get

( )

Step 7. For each n =1, 2, compute P1,^*_n

( )

0 using (3.37) in terms of P . _0,0

Our analysis is based on the following system performance measures of the F policy G/M/1/K queue with exponential startup time. Let

L ≡ the average number of customers in the system; s

P ≡ the probability that the server is busy; b

P ≡ the probability that the server requires a startup time before starting the s

service;

P ≡ the probability that the server is blocked. bl

The expressions for L , _s P , _b P , and _s P are give by _bl

We derive the total expected cost function per unit time for the F policy

G/M/1/K queue with startup times, in which F is a decision variable. The main purpose of this subsection is to determine the optimum operating F policy so as to minimize this total expected cost function. Let

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

C ≡ busy cost per unit time for a busy server; b

C ≡ startup cost per unit time for the preparatory work of the server before s

starting the service;

C ≡ fixed cost for every lost customer when the system is blocked. bl

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

( ) _{h s b b s s bl bl}

TC F =C L +C P +C P C+

λ

P . (3.58)

The optimal value of F , F is determined by satisfying the following ^* inequality

* *

( 1) ( )

TC F − ≥TC F and TC F( ^*+ ≥1) TC F( ^*). (3.59)

3.5.2 Numerical examples

We set the system capacity K=15. We perform a sensitivity analysis for changes in the optimum value F along with changes in specific values of the ^* system parameters. We consider three simple examples for three different interarrival time distributions such as exponential, 3-stage Erlang, and deterministic.

The following cost elements are employed:

Case 1: C_h =10, C_b =200, C_s =250, C_bl =350.

Case 2: C_h =10, C_b =200, C_s =250, C_bl =400.

Case 3: C_h =10, C_b =200, C_s =300, C_bl =400.

Case 4: C_h =10, C_b =225, C_s =300, C_bl =400.

Case 5: C_h =15, C_b =225, C_s =300, C_bl =400.

In this subsection, we provide the numerical results of the optimal value F ^* and the minimum expected cost for three interarrival time distributions and specific values of

λ

,

μ

,

β

. We first fix

( ^{μ β}

^,

)

= (1.0, 3.0) and choose different values of λ = 0.55, 0.65, 0.75. Next, we fix

( ^{λ β}

^,

)

= (0.7, 3.0) and consider various values of

μ

= 1.0, 1.1, 1.2. Finally, we fix

( ^{λ μ}

^,

)

= (0.7, 1.0) and select different values of

β

= 2.0, 4.0, 5.0.

The optimal value of F, F , and its minimum expected cost ^{* TC F}

( )

^{* for the}

above five cases are shown in Tables 4-6. For fixed values of

( ^{μ β}

^,

)

and various values of λ in Tables 4-6, we observe that (i) ^{TC F}

( )

^* increases as λ increases for any case; and (ii) F decreases as ^*

λ

increases for any case. For fixed values of

( ^{λ β}

^,

)

and various values of

μ

in Tables 4-6, we find that (i) ^{TC F}

( )

^*

decreases as

μ

increases for any case; and (ii) F increases as ^*

μ

increases for any case. Again, for fixed

( ^{λ μ}

^,

)

and various values of

β

in Tables 4-6, we observe that (i) ^{TC F}

( )

^* slightly decreases as

β

increases for any case; and (ii)

F does not change at all when *

β

changes from 2.0 to 5.0 for any case.

Intuitively, F is insensitive to changes in ^*

β

.

It can be easily seen from Tables 4 through 6 that (i) F increases as ^* C _h decreases (see cases 4-5); and (ii) C has a larger effect on _h F than ^* C , _b C and _s C (see cases 3-4, cases 2-3 and cases 1-2). bl

Table 4. The optimal value of F and its minimum expected cost for exponential interarrival time

λ (^{μ β,} ) (^{= 1.0,3.0}) ^μ (^{λ β,} ) (^{= 0.7,3.0}) ^β (^{λ μ,} ) (^{= 0.7,1.0})

0.55 0.65 0.75 1.0 1.1 1.2 2.0 4.0 5.0

Case1 F^* 6 4 3 4 6 8 4 4 4

( *)

TC F 122.209 148.361 177.914 162.635 144.660 130.652 162.654 162.626 162.620

Case2 F^* 8 6 5 5 8 11 5 5 5

( *)

TC F 122.215 148.425 178.303 162.803 144.705 130.663 162.823 162.793 162.787

Case3 F^* 8 6 5 5 8 11 5 5 5

( *)

TC F 122.216 148.428 178.318 162.810 144.708 130.664 162.834 162.798 162.791

Case4 F^* 7 5 4 4 7 10 5 4 4

( *)

TC F 135.963 164.647 196.879 180.230 160.598 145.243 180.253 180.218 180.211

Case5 F^* 4 3 2 2 4 6 2 2 2

( *)

TC F 142.056 173.713 210.174 191.254 169.196 152.207 191.276 191.242 191.235

Table 5. The optimal value of F and its minimum expected cost for 3-stage Erlang interarrival time

Table 5. The optimal value of F and its minimum expected cost for 3-stage Erlang interarrival time

在文檔中 M/G/1/K與G/M/1/K排隊含啟動時間的F方策與N方策之相互關係 (頁 17-0)