The Optimal T Policy - Optimization of the T Policy M/G/1 Queue with Server Breakdowns and

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

4.3 The Optimal T Policy

We develop the total expected cost function per unit time for the policy M/G/1 queue with server breakdowns and general startup times in which is a decision variable. We determine the optimum value of the control parameter for our constructing expected cost function. Let us define the cost elements as follows:

T T

T

C

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

C

s ≡ setup cost 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.

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

( ) 1

Omitting these cost terms are not functions of the decision variable T , the optimization problem in (4.19) is equivalent to minimize the following equation:

2 2

4.4 Sensitivity Analysis

A system analyst is concerned with varying the system parameters over a reasonable range and observing the relative change in the system performance measures. A sensitivity investigation of different system parameters (λ ,

μ

α

β

γ

) and cost parameters (

C

_h,

C

_s,

C

_i,

C

_sp,) levels is particularly valuable when evaluating future conditions, where

μ

_S =1

μ

_R =1

β

and

μ

_U =1

γ

. We can assess how robust the results are to system input parameters. In the following, we conduct some sensitivity investigations on the optimal value based on changes in the values of system parameters and cost parameters. From (4.22), differentiating with respect to

It follows from (4.25) that

T

^* increases in

μ

. Similarly, differentiating with respect to we can see how

γ

affects while startup time distribution is given. For special case, suppose that the startup time distribution obeys an exponential distribution with

T

mean

μ

_U =1

γ

. Substituting σ_U²

=

μ_U² into (4.22) and then differentiating with

We present some numerical computations to verify the analytical results, and show how to make the decision based on minimizing the cost function . The sensitivity investigation concentrates mainly on the exponential startup time distribution. The cost parameters

F T

( )

s 1000

C

= ,

C

_h = , 5

C

_sp

= 100

, are fixed.

We consider the following four cases.

i 60 quickly and the values of rarely change for different values of

T

optimal value, , and the corresponding minimum expected cost are displayed in Table 4.1 for parameters

T

F T

( )^∗

= 0.05

β

= , and 3

γ

= . 3

It appears from Figure 4.3 that (i)

T

decreases in α

；(ii) as

α is fixed, the

larger

β has larger T

^*; and (iii)

T

has an upper bound as α closes to zero.

Figure 4.4 reveals that (i)

T

increases in β but T

is insensitive to β as β is

large; and (ii) as

β is fixed, the larger α has the smaller

. Furthermore, the optimal value, , and the corresponding minimum expected cost are shown in Table 4.2 for parameters

T

F T

( )^∗

= 0.5

μ

= , and 1

γ

= . 3

Figure 4.5 indicates that (i)

T

^* increases in

γ

; and (ii) as

γ

is smaller than 0.4 ,

T

^* increases quickly but

T

is insensitive to γ as γ is larger than 0.4. The

optimal value, , and the corresponding minimum expected cost are displayed in Table 4.3 for parameters

T

F T

( )^∗

= 0.3

μ

= , 1 α

= 0.05

, and

γ

= . 3

To see how

T

changes when the cost parameter changes, we set

= 0.3

μ

= , 1 α

= 0.5

β

= , 3

γ

= , choose 3

C

_sp

= 100

C

_i =60, and vary the specified values of ( ,

C C

_s _h). We observe from Table 4.4 that

T

increases in C and

_s decreases in

C

_h. On the other hand, we select

C

_s =1000,

C

_h = , and change the 5 specified values of

( C

_sp

, ) C

_i . Table 4.5 reveals that

T

increases in C

_sp

and

decreases in

C

_i, but

T

is insensitive to ( C

_sp

, ) C

_i .

Figure 4.1 Plots of (

λ

, T^*) 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

T

λ

Figure 4.2 Plots of (

μ

, T^*) 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

T

μ

Table 4.1. The optimal and minimum expected cost with various values of

T

F T

( ^*)

( , )

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) T* 22.6241 30.3119 32.4767 33.5071 39.8499 24.1892 15.9017 9.3975

( *)

F T 57.8362 87.6679 97.0150 101.6107 87.9832 84.6451 72.1052 50.1233

Figure 4.3 Plots of (

α

, T^*) with

λ

=0.5, μ =1, β =1, 2, 3, 4, γ =3,

C =1000,

h=5,

C C

_sp= 100,

C

_i=60

T

Figure 4.4 Plots of (

β

, T^*) 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

T

Table 4.2. The optimal and minimum expected cost with various values of

T

F T

( ^*)

( , )

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)

T* 6.7925 8.3908 8.8621 9.0889 8.6765 7.4712 6.0420 4.1808

( *)

F T 59.4781 76.9783 82.4260 85.0945 80.2654 66.7251 51.7803 34.1213

Figure 4.5 Plots of (

γ

, T^*) with

λ

=0.3, μ =1,

α

=0.05, β =3,

C =1000,

C

_h=5,

C

sp= 100,

C

_i=60

T

Table 4 3. The optimal and minimum expected cost with various values of

T

F T

( ^*)

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 2.4 2.8 3.2

T* 7.3990 8.1910 8.4873 8.6419 8.7368 8.8009 8.8471 8.8821

( *)

F T 126.8279 126.2007 125.7975 125.5548 125.3955 125.2834 125.2006 125.1368

Table 4.4. The optimal and minimum expected cost with various values of

T

F T

( ^*)

(

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)

T* 8.8657 6.1757 4.9847 4.2753 3.8317 4.7349 5.4999 5.8470

( *)

F T 125.1667 159.6880 186.1419 208.4142 116.9593 133.4406 147.3823 153.7043

Table 4.5. The optimal and minimum expected cost with various values of

T

F T

( ^*)

( C

_sp

, C

)

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)}

T* 8.8959 8.8808 8.8657 8.8506 8.8203 8.8355 8.8506 8.8657

( *)

F T 97.6411 104.4540 111.2667 118.0792 100.4285 100.5665 100.7042 100.8417

Chapter 5 Maximum Entropy Analysis to the T Policy M/G/1 Queue with Server Breakdowns and Startup Times

In this chapter, we use the maximum entropy approach to solve the steady-state probabilities of the policy M/G/1 queue with server breakdowns and general startup time. Besides the constraints of normalizing condition and the probability of the various server statuses, the maximum entropy solutions are used to derive the queue length distributions using the first moment and second moment of the number of customers in the system, respectively. We derive the approximate formulas for the steady-state probability distributions of the queue length and perform a comparative analysis between the approximate results with established exact results for various distributions, such as exponential (M), k-stage Erlang (E

T

k), and deterministic (D). The experiment demonstrates that the maximum entropy approach is accurate enough for practical purposes and is a useful method for solving complex queueing systems by using the first moment of the number of customers in the system, which the use is better than the second moment of the number of customers in the system.

5.1 Assumptions and Notations

We consider a policy M/G/1 queue in which the server performs a startup before providing service and is typically subject to unpredictable breakdowns. As soon as the system is empty, the server is immediately turned off. After the time is elapsed length , the server is immediately turned on but is temporarily unavailable to serve the 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.

T

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. The server is turned off when the system is empty. If some customers are accumulated in the queue after unit time is elapsed since system empty, the server is immediately turned on (i.e.

begin startup) but is temporarily unavailable for 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 ( ) with a finite mean

T

U( )

F t t

≥0

μ

U and a finite variance σ . Once the startup is terminated, the server _U² 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 exact steady-state solutions to the

T

policy M/G/1 queue with service times, repair times or startup times distribution of the general type have not been found. It is extremely difficult, if not impossible, to obtain the explicit formulas such as the steady-state probability mass function of the number of customers and the expected waiting time for the policy M/G/1 queue in which the repair times and startup times are generally distributed. However, one can utilize the maximum entropy principle to approximate the

T

policy M/G/1 queue with general repair times and general startup times. This becomes particularly helpful when some system performance measures (for instance, the expected number of customers in the system, the probability that the server is busy, broken down, etc) are known. In this paper, we utilize the maximum entropy principle associated with five basic known results from the literature to study the

T

policy M/G/1 queue with general repair times and general startup times. Next, we replace the first moment of customers in the system by the second moment of the queue length to study the policy M/G/1 queue with general repair times and general startup times

T

The purpose of this chapter is:

(ii) to present some important system performance measures for the

T

policy M/G/1 queue with general repair times and general startup times;

(iii) to develop the maximum entropy (approximate) solutions for the

^T

policy M/G/1 queue with general repair times and general startup times by using Lagrange’s method;

(iv) to obtain approximate results for the expected waiting time in the queue;

(v) to perform a comparative analysis between exact results and two approximate results obtained through maximum entropy principle by using (i) the first moment of the queue length and (ii) the second moment of the queue length.

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

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 ( )

G z −

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

( )

A t −

number of customers arriving in 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 startup period

( )

f ⋅ −

Laplace-Stieltjes transform (LST) of startup time

( )

W z −

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

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

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

T

L

2T − second moment of the queue length in the policy M/G/1 queue with server breakdowns and general startup times

T

is turned on and working

n

2( )

P

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

n

W

−

approximate expected waiting time in the queue

5.2 The 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 results of Medhi and Templeton [34], the probability generating function (p.g.f.) of the number of customers in the ordinary M/G/1 queue with server breakdowns is given by

(1 )(1 ) ( )

2 2

(

μ_S

+

σ_S

) +

αμ μ_S

(

_R²

+

σ_R²

)

. The traffic intensity

ρ

_H is assumed to be less than 1. We consider a Poisson arriving process. Let

ξ

_i denote the elapsed time between the (i-1)st and the i-th customer arriving. Following Ross [36], the event

ξ

_i are i.i.d.

exponential random variables with mean 1

λ

. Let

A t ( )

denote the number of

Let be the p.g.f. of the number of customers waiting in the queue during idle period

Let be the p.g.f. of the number of customers arriving during startup period . Then we get

G

U

(1 )

( )

^z ^t

G

z = e

^{− −} ^λ and the Laplace-Stieltjes transform (abbreviated LST) of

G

_U( )

z

given by

[ ]

0^∞

G

_U( )

z dF t

_U( )=

f

λ

(1−

z

)

∫

Because the Poisson process from any point on arrival is independent of all that has previously occurred,

W z ( ) = G z f

( )

_U[(1−

z

) ]

λ = e

^{− −}⁽¹ ^z⁾^λ^T

f

_U[(1−

z

) ]

λ

. For the

T

policy M/G/1 queue with server breakdowns and startup time, we get the idle period plus the startup period as . Using the well-known decomposition property concerning M/G/1 vacation queue of Fuhrmann and Cooper [14], we obtain the p.g.f.

of number of customers in the policy M/G/1 queue with server breakdowns and general startup times as follows:

( )

where is given in (5.1). Let and denote the first moment and the second moment of the number of customers in the

T

policy M/G/1 queue with server breakdowns and general startup times, respectively. Thus we obtain

( )

5.3 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

T

0,_I

( ) ≡

P n

probability that there are customers in the system when the server is turned off, where .

n 0, 1, 2, n = "

0,_U

( ) ≡

P n

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

n 1, 2, 3, n = "

1( )≡

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

( )

2 2

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 policy M/G/1 queue with server breakdowns and general startup times. The maximum entropy solutions are obtained by maximizing (5.5) subject to the following five constraints, written as,

T

(i) normalizing condition

, (5.6)

(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

, (5.10)

where

τ

₁-

τ

₅ are the Lagrangian multipliers corresponding to constraints (5.6)-(5.10), respectively.

5.3.1 The maximum entropy solutions with the first moment of the queue length

To get the maximum entropy solutions, , , , maximizing in (5.5) subject to constraints (5.6)-(5.10) is equivalent to maximizing (5.11).

0,_U

( )

P n P n

₁( )

P n

₂( )

The maximum entropy solutions are obtained by taking the partial derivatives of with respect to , and (

i

= 1, 2), and setting the results equal to zero, namely,

y

P

_0,_I

( ) n P

_0,_U

( ) n P n

_i( )

It implies from (5.12)-(5.16) that we obtain

(1 1)

. (5.21)

Substituting (5.23)-(5.25) into (5.7)-(5.9), respectively, yields

S

ubstituting (5.22)-(5.26) into (5.6) and using (5.27)-(5.29), we obtain

( )

Substituting (5.22)-(5.26) and (5.30) into (5.10) and taking the algebraic manipulations, we obtain

( )

5 5

5.3.2 The maximum entropy solutions with the second moment of the queue length

In this section, we develop another maximum entropy solutions based on the second moment of the queue length. The maximum entropy is obtained by maximizing (5.4) by replacing the (5.10) with the second moment of the number of customers in the system. That is, maximize the Lagrangian function

y

₂ given by

2 0, 0, 0, 0,

The maximum entropy solutions are obtained by taking the partial derivatives of with respect to , and (

i

= 1, 2), and setting the results equal

2 2

Substituting (5.46)-(5.48) into (5.7)-(5.9), respectively, yields

( )

Substituting (5.44)-(5.48) into (5.6) and then replacing by (5.49)-(5.51), we get

Substituting (5.44)-(5.48) and (5.52) into (5.37) and taking the algebraic manipulations, we obtain

( )

approximate solution of

ϕ

₅ is obtained using Newton method. In iteration process, one requires the successive approximate solution of

ϕ

₅ is less than 0.00001.

5.4 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.

T

5.4.1 The exact expected waiting time in the queue

Let denote the exact expected waiting time in the queue. Using (5.3) and Little’s formula, we obtain

W

5.4.2 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 greater than or equal to 0.

(ii) Startup state denoted by : the server begins startup and the number of customers waiting in the system is greater than or equal to 1.

U

(iii)Busy state denoted by

B : the server is busy and provides service to a

_S customer.

(iv) Repair state denoted by

R : the server is broken down and being repaired.

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

C

one of the states

I ,

U

_S,

B and

R are described, respectively, as follows:

_S we have

(i) In idle state

I : The server will begin startup after customer

_S arrive and n customers in front of him waiting for service. Following the well known theorem of renewal theory (mean residual life), the mean remaining idle time is

C

(ii) In startup state : Using the same argument as (i), we get the mean remaining startup time

U

[

] 2

E U

μ . Thus we obtain the expected waiting time of customer at the startup state is

C n

μ_S

+ E U [

] 2

μ_U .

(iii)In busy state

B : Since the server is turned on and working, customer

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

C n

C n μ

_S.

(iv) In repair state

R : Using the same argument as (ii),we have the expected waiting

_S time of customer

C

at the repair state is

n

μ_S

+ E R [

] 2

μ_R.

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

The primary objective of this section is to examine the accuracy of the two maximum entropy results. We present specific numerical comparisons between the exact results and the two 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 queue as the

T

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.

T

This section includes the following three subsections:

(i) comparative analysis for the

T

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

(ii) comparative analysis for the

T

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

(iii)comparative analysis for the

T

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

5.5.1 Comparative analysis for the T policy M/M(M,M)/1 and M/D(D,D)/1 queues

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

T

policy M/M(M,M)/1 queues, we obtain

W

T

, 1

μ

_S =

E

[

S

²]=2

μ

²,

E

[

S

³]=6

μ

³, μ_R

= 1

, E

[

R

²]=2

β

²,

E

[

R

³]=6

β

³, ,

γ

μ

_U =

E

[

U

²]=2

γ

², and

E U

[ ³]=6

γ

³ . For the

T

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

μ

_S =1

μ

E

[

S

²]=1

μ

²,

E

[

S

³]=1

μ

³, μ_R

= 1

, E

[

R

²]=1

β

²,

, 1 ]

[

R

³ =

β

E μ

_U =1

γ

E U

[ ²] 1=

γ

² and

E U

[ ³] 1=

γ

³.

We set

T

=5 and

T

=10, and choose the various values of

λ

,μ ,

α

,β , and γ . 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 0.2 to 0.8.

Case 2: We fix

λ

=0.3,

α

=0.05, β =3.0, γ =3.0, and vary the values of μ from 0.5 to 2.0.

Case 3: We fix

λ

=0.3, μ =1.0, β =3.0, γ =3.0, and vary the values of

α

from 0.05 to 0.2.

Case 4: We fix

λ

=0.3, μ =1.0,

α

=0.05, γ =3.0, and vary the values of β from 2.0 to 5.0.

Case 5: We fix

λ

=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 of and for the policy M/M(M,M)/1 and M/D(D,D)/1 queues are shown in Table 5.1 for the above five cases using the constraint of the first moment of the queue length. The most relative error percentages are small (0.2-6.0%). Table 5.2 presents numerical results using the constraint of the second moment of the queue length. The range of relative error percentages is wider (1.3-27.9%).

W

_q^*

T

5.5.2 Comparative analysis for the T policy M/E

(E

,E

)/1 and M/M(E

,E

)/1 queues

Here we perform a comparative analysis between the exact and the two approximate (maximum entropy) for the

T

policy M/E₃(E₄,E₃)/1 and M/M(E₃, M/M(E3,E2)/1 queues are shown in Table 5.3 for the above five cases based on the constraint of the first moment of the queue length. The most relative error percentages are very small (0.2-4.3%). Table 5.4 displays numerical results using the constraint of the second moment of the queue length. The relative error percentages are lager.

W

_q^*

T

5.5.3 Comparative analysis for the T policy M/E

(E

,D)/1 and M/E

(E

,M)/1 queues

Here we perform a comparative analysis between the exact and the two approximate (maximum entropy) for the

T

policy M/E₃(E₄,D)/1 and M/E₃(E₄, first moment of queue length. Again, the most relative error percentages are very small (0.2-4.4%). Table 5.6 shows numerical results using the constraint of the second moment of the queue length. The relative error percentages are lager (1.1-26.8%).

W

_q^*

T

Table 5.1. Comparison of exact and approximate for the T policy M/M(M,M)/1 and M/D(D,D)/1 queues with the first moment of the queue length.

W

_q^*

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

T

=10

T

=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 2.9380 2.9248 0.4487 5.4329 5.4114 0.3957 2.7971 2.8878 3.2408 5.2971 5.3794 1.5540 0.4 3.3776 3.3483 0.8673 5.8726 5.8267 0.7821 3.0169 3.1968 5.9607 5.5169 5.6801 2.9575 0.6 4.2758 4.2229 1.2378 6.7708 6.6929 1.1502 3.4660 3.7313 7.6530 5.9660 6.2063 4.0271 0.8 7.1307 7.0220 1.5234 9.6256 9.4837 1.4742 4.8935 5.2281 6.8386 7.3935 7.6948 4.0754

μ Case 2. (λ=⁰^.³^, α=⁰^.⁰⁵^, β=³^.⁰^, γ =³^.⁰)

0.5 5.8660 5.7870 1.3466 8.3609 8.2570 1.2432 4.2611 4.8183 13.0769 6.7611 7.2933 7.8718 1.0 3.1256 3.1049 0.6626 5.6206 5.5874 0.5904 2.8909 3.0265 4.6872 5.3909 5.5140 2.2817 1.5 2.8515 2.8397 0.4126 5.3464 5.3263 0.3756 2.7539 2.8113 2.0868 5.2539 5.3030 0.9352

2.0 2.7695 2.7613 0.2959 5.2645 5.2501 0.2741 2.7129 2.7436 1.1319 5.2129 5.2374 0.4692 α ^{Case 3.} (λ=0.3, μ=1.0, β=3.0, γ =3.0)

0.05 3.1256 3.1049 0.6626 5.6206 5.5874 0.5904 2.8909 3.0265 4.6872 5.3909 5.5140 2.2817 0.10 3.1462 3.1044 1.3284 5.6411 5.5744 1.1832 2.9012 3.0220 4.1645 5.4012 5.4970 1.7741 0.15 3.1672 3.104 1.9973 5.6622 5.5615 1.7782 2.9117 3.0177 3.6387 5.4117 5.4802 1.2648

0.20 3.1888 3.1037 2.669 5.6838 5.5488 2.3753 2.9225 3.0134 3.1097 5.4225 5.4634 0.7539 β Case 4. (λ=0.3, μ=1.0, α=0.05, γ =3.0)

2.0 3.1376 3.1064 0.9950 5.6326 5.5827 0.8865 2.8969 3.0251 4.4246 5.3969 5.5064 2.0276 3.0 3.1256 3.1049 0.6626 5.6206 5.5874 0.5904 2.8909 3.0265 4.6872 5.3909 5.5140 2.2817 4.0 3.1201 3.1047 0.4967 5.6151 5.5903 0.4426 2.8882 3.0273 4.8177 5.3882 5.5180 2.4084 5.0 3.1170 3.1046 0.3972 5.6120 5.5921 0.3540 2.8866 3.0280 4.8958 5.3866 5.5205 2.4844

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

2.0 3.2213 3.2001 0.6578 5.7105 5.6768 0.5890 2.9743 3.1094 4.5418 5.4743 5.5969 2.2393 3.0 3.1256 3.1049 0.6626 5.6206 5.5874 0.5904 2.8909 3.0265 4.6872 5.3909 5.5140 2.2817 4.0 3.0795 3.0590 0.6651 5.5766 5.5436 0.5912 2.8493 2.9850 4.7630 5.3493 5.4725 2.3033 5.0 3.0524 3.0321 0.6665 5.5505 5.5177 0.5916 2.8243 2.9601 4.8096 5.3243 5.4476 2.3165

Table 5.2. Comparison of exact and approximate for the T policy M/M(M,M)/1 and M/D(D,D)/1 queues with the second moment of the queue length.

W

_q^*

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

T

=10

T

=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 2.9380 2.4113 17.9259 5.4329 4.3457 20.0120 2.7971 2.3352 16.5134 5.2971 4.2826 19.1522 0.4 3.3776 3.1851 5.7009 5.8726 5.3318 9.2091 3.0169 2.9650 1.7229 5.5169 5.1444 6.7537 0.6 4.2758 4.2200 1.3039 6.7708 6.3699 5.9205 3.4660 3.6093 4.1346 5.9660 5.8317 2.2516 0.8 7.1307 7.2423 1.5663 9.6256 9.2248 4.1640 4.8935 5.1280 4.7924 7.3935 7.2667 1.7141

μ Case 2. (λ=⁰^.³^, α=⁰^.⁰⁵^, β=³^.⁰^, γ =³^.⁰)

0.5 5.8660 6.1944 5.5987 8.3609 8.2019 1.9021 4.2611 4.8731 14.3622 6.7611 6.9833 3.2869 1.0 3.1256 2.8071 10.1918 5.6206 4.8890 13.0167 2.8909 2.6753 7.4599 5.3909 4.7784 11.3621 1.5 2.8515 2.3752 16.7033 5.3464 4.4470 16.8223 2.7539 2.3247 15.5840 5.2539 4.4079 16.1013

2.0 2.7695 2.2199 19.8454 5.2645 4.2831 18.6423 2.7129 2.1894 19.2948 5.2129 4.2612 18.2561 α ^{Case 3.} (λ=0.3, μ=1.0, β=3.0, γ =3.0)

0.05 3.1256 2.8071 10.1918 5.6206 4.8890 13.0167 2.8909 2.6753 7.4599 5.3909 4.7784 11.3621 0.10 3.1462 2.8136 10.5705 5.6411 4.8845 13.4122 2.9012 2.6760 7.7622 5.4012 4.7686 11.7118 0.15 3.1672 2.8203 10.9541 5.6622 4.8802 13.8099 2.9117 2.6768 8.0686 5.4117 4.7589 12.0632

0.20 3.1888 2.8272 11.3426 5.6838 4.8761 14.2100 2.9225 2.6777 8.3789 5.4225 4.7493 12.4165 β Case 4. (λ=0.3, μ=1.0, α=0.05, γ =3.0)

2.0 3.1376 2.3125 26.2994 5.6326 4.0656 27.8209 2.8969 2.6767 7.6015 5.3969 4.7745 11.5324 3.0 3.1256 2.3076 26.1719 5.6206 4.0649 27.6784 2.8909 2.6753 7.4599 5.3909 4.7784 11.3621 4.0 3.1201 2.3057 26.1037 5.6151 4.0651 27.6049 2.8882 2.6748 7.3872 5.3882 4.7806 11.2760 5.0 3.1170 2.3047 26.0615 5.6120 4.0653 27.5601 2.8866 2.6747 7.3429 5.3866 4.7820 11.2241

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

2.0 3.2213 2.8870 10.3778 5.7105 4.9703 12.9622 2.9743 2.7411 7.8406 5.4743 4.8522 11.3630 3.0 3.1256 2.8071 10.1918 5.6206 4.8890 13.0167 2.8909 2.6753 7.4599 5.3909 4.7784 11.3621 4.0 3.0795 2.7698 10.0576 5.5766 4.8497 13.0346 2.8493 2.6431 7.2362 5.3493 4.7418 11.3569 5.0 3.0524 2.7483 9.9629 5.5505 4.8266 13.0422 2.8243 2.6240 7.0903 5.3243 4.7199 11.3521

Table 5.3. Comparison of exact W

and approximate W

_q^*

for the T policy M/E

(E

,E

)/1 and M/M(E

,E

)/1 queues with the first moment of the queue length.

M/E₃(E₄,E₃)/1 M/M(E₃,E₂)/1

T

=10

T

=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 2.8440 2.9000 1.9704 5.3423 5.3900 0.8931 2.9323 2.9191 0.4489 5.4298 5.4083 0.3957 0.4 3.1370 3.2471 3.5104 5.6353 5.7288 1.6586 3.3712 3.3419 0.8677 5.8687 5.8228 0.7822 0.6 3.7356 3.8948 4.2617 6.2339 6.3681 2.1530 4.2677 4.2149 1.2382 6.7652 6.6874 1.1503 0.8 5.6382 5.8251 3.3149 8.1365 8.2901 1.8876 7.1175 7.0091 1.5238 9.6150 9.4732 1.4743

μ Case 2. (λ=⁰^.³^, α=⁰^.⁰⁵^, β=³^.⁰^, γ =³^.⁰)

0.5 4.7957 5.1409 7.1971 7.2940 7.6142 4.3895 5.8579 5.7790 1.3471 8.3554 8.2515 1.2434 1.0 2.9691 3.0525 2.8100 5.4674 5.5383 1.2975 3.1196 3.0990 0.6629 5.6171 5.5839 0.5905 1.5 2.7863 2.8207 1.2343 5.2847 5.3107 0.4932 2.8458 2.8340 0.4127 5.3433 5.3232 0.3756

2.0 2.7317 2.7495 0.6494 5.2301 5.2415 0.2198 2.7640 2.7558 0.2960 5.2615 5.2471 0.2742 α ^{Case 3.} (λ=0.3, μ=1.0, β=3.0, γ =3.0)

0.05 2.9691 3.0525 2.8100 5.4674 5.5383 1.2975 3.1196 3.0990 0.6629 5.6171 5.5839 0.5905 0.10 2.9827 3.0493 2.2335 5.4810 5.5226 0.7596 3.1393 3.0976 1.3292 5.6368 5.5701 1.1833 0.15 2.9966 3.0461 1.6536 5.4949 5.5070 0.2198 3.1596 3.0964 1.9985 5.6571 5.5565 1.7784

0.20 3.0109 3.0431 1.0703 5.5092 5.4915 0.3220 3.1804 3.0954 2.6708 5.6779 5.5430 2.3757 β Case 4. (λ=0.3, μ=1.0, α=0.05, γ =3.0)

2.0 2.9770 3.0520 2.5210 5.4753 5.5316 1.0285 3.1306 3.0995 0.9956 5.6281 5.5782 0.8866 3.0 2.9691 3.0525 2.8100 5.4674 5.5383 1.2975 3.1196 3.0990 0.6629 5.6171 5.5839 0.5905 4.0 2.9655 3.0531 2.9540 5.4638 5.5420 1.4318 3.1145 3.0990 0.4969 5.6120 5.5871 0.4427 5.0 2.9634 3.0535 3.0401 5.4617 5.5443 1.5123 3.1115 3.0992 0.3974 5.6090 5.5891 0.3540

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

2.0 3.0565 3.1395 2.7154 5.5529 5.6234 1.2698 3.2091 3.188 0.6584 5.7037 5.6701 0.5891 3.0 2.9691 3.0525 2.8100 5.4674 5.5383 1.2975 3.1196 3.099 0.6629 5.6171 5.5839 0.5905 4.0 2.9259 3.0096 2.8589 5.4250 5.4961 1.3116 3.0757 3.0553 0.6653 5.5743 5.5413 0.5912 5.0 2.9002 2.9840 2.8886 5.3996 5.4709 1.3201 3.0497 3.0294 0.6667 5.5487 5.5159 0.5916

Table 5.4. Comparison of exact W

and approximate W

_q^*

for the T policy M/E

(E

,E

)/1 and M/M(E

,E

)/1 queues with the second moment of the queue length.

M/E₃(E₄,E₃)/1 M/M(E₃,E₂)/1

T

=10

T

=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 2.8440 2.3601 17.0161 5.3423 4.3031 19.4523 2.9323 2.4047 17.9936 5.4298 4.3419 20.0361 0.4 3.1370 3.0358 3.2263 5.6353 5.2046 7.6445 3.3712 3.1777 5.7384 5.8687 5.3272 9.2262 0.6 3.7356 3.8050 1.8578 6.2339 6.0034 3.6981 4.2677 4.2111 1.3262 6.7652 6.3637 5.9353 0.8 5.6382 5.8136 3.1111 8.1365 7.8945 2.9743 7.1175 7.2283 1.5561 9.6150 9.2132 4.1789

μ Case 2. (λ=⁰^.³^, α=⁰^.⁰⁵^, β=³^.⁰^, γ =³^.⁰)

0.5 4.7957 5.3012 10.5402 7.2940 7.3744 1.1016 5.8579 6.1859 5.5987 8.3554 8.1957 1.9111 1.0 2.9691 2.7180 8.4575 5.4674 4.8141 11.9485 3.1196 2.8001 10.2412 5.6171 4.8848 13.0365 1.5 2.7863 2.3412 15.9772 5.2847 4.4206 16.3494 2.8458 2.3685 16.7732 5.3433 4.4432 16.8444

2.0 2.7317 2.1994 19.4860 5.2301 4.2684 18.3877 2.7640 2.2133 19.9243 5.2615 4.2794 18.6653 α ^{Case 3.} (λ=0.3, μ=1.0, β=3.0, γ =3.0)

0.05 2.9691 2.7180 8.4575 5.4674 4.8141 11.9485 3.1196 2.8001 10.2412 5.6171 4.8848 13.0365 0.10 2.9827 2.7204 8.7912 5.4810 4.8059 12.3160 3.1393 2.8057 10.6280 5.6368 4.8795 13.4361 0.15 2.9966 2.7230 9.1293 5.4949 4.7979 12.6855 3.1596 2.8114 11.0196 5.6571 4.8742 13.8381

0.20 3.0109 2.7257 9.4719 5.5092 4.7899 13.0571 3.1804 2.8173 11.4159 5.6779 4.8692 14.2423 β Case 4. (λ=0.3, μ=1.0, α=0.05, γ =3.0)

2.0 2.9770 2.2502 24.4134 5.4753 4.0131 26.7055 3.1306 2.3054 26.3590 5.6281 4.0610 27.8446 3.0 2.9691 2.2475 24.3020 5.4674 4.0145 26.5736 3.1196 2.3016 26.2212 5.6171 4.0614 27.6967 4.0 2.9655 2.2465 24.2439 5.4638 4.0155 26.5065 3.1145 2.3001 26.1495 5.6120 4.0619 27.6214 5.0 2.9634 2.2460 24.2082 5.4617 4.0162 26.4658 3.1115 2.2992 26.1057 5.6090 4.0623 27.5758

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

2.0 3.0565 2.7885 8.7688 5.5529 4.8904 11.9301 3.2091 2.8729 10.4768 5.7037 4.9622 13.0004 3.0 2.9691 2.7180 8.4575 5.4674 4.8141 11.9485 3.1196 2.8001 10.2412 5.6171 4.8848 13.0365 4.0 2.9259 2.6841 8.2662 5.4250 4.7766 11.9514 3.0757 2.7654 10.0896 5.5743 4.8470 13.0478 5.0 2.9002 2.6642 8.1389 5.3996 4.7543 11.9511 3.0497 2.7451 9.9868 5.5487 4.8245 13.0523

Table 5.5. Comparison of exact W

and approximate W

_q^*

for the T policy M/E

(E

,D)/1 and M/E

(E

,M)/1 queues with the first moment of the queue length.

M/E₃(E₄,D)/1 M/E₃(E₄,M)/1

T

=10

T

=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 2.8405 2.8966 1.9732 5.3405 5.3882 0.8935 2.8509 2.9070 1.9648 5.3459 5.3936 0.8923 0.4 3.1336 3.2437 3.5151 5.6336 5.7270 1.6593 3.1440 3.2540 3.5012 5.6389 5.7324 1.6571 0.6 3.7321 3.8914 4.2666 6.2321 6.3664 2.1539 3.7425 3.9017 4.2519 6.2375 6.3717 2.1512 0.8 5.6347 5.8217 3.3177 8.1347 8.2883 1.8883 5.6451 5.8319 3.3091 8.1401 8.2936 1.8862

μ Case 2. (λ=⁰^.³^, α=⁰^.⁰⁵^, β=³^.⁰^, γ =³^.⁰)

0.5 4.7922 5.1374 7.2031 7.2922 7.6124 4.3908 4.8027 5.1477 7.1853 7.2976 7.6177 4.3868 1.0 2.9656 3.0491 2.8139 5.4656 5.5366 1.2981 2.9760 3.0594 2.8023 5.4710 5.5419 1.2963 1.5 2.7829 2.8173 1.2362 5.2829 5.3089 0.4935 2.7933 2.8276 1.2304 5.2882 5.3143 0.4926

2.0 2.7283 2.7460 0.6506 5.2283 5.2398 0.2199 2.7387 2.7564 0.6471 5.2336 5.2451 0.2195 α ^{Case 3.} (λ=0.3, μ=1.0, β=3.0, γ =3.0)

0.05 2.9656 3.0491 2.8139 5.4656 5.5366 1.2981 2.9760 3.0594 2.8023 5.4710 5.5419 1.2963 0.10 2.9792 3.0458 2.2373 5.4792 5.5208 0.7602 2.9896 3.0562 2.2260 5.4846 5.5262 0.7585 0.15 2.9931 3.0427 1.6572 5.4931 5.5052 0.2203 3.0035 3.0530 1.6463 5.4985 5.5105 0.2186

0.20 3.0074 3.0397 1.0739 5.5074 5.4897 0.3215 3.0179 3.0499 1.0633 5.5128 5.4950 0.3231 β Case 4. (λ=0.3, μ=1.0, α=0.05, γ =3.0)

2.0 2.9735 3.0486 2.5248 5.4735 5.5298 1.0290 2.9839 3.0589 2.5133 5.4789 5.5351 1.0273 3.0 2.9656 3.0491 2.8139 5.4656 5.5366 1.2981 2.9760 3.0594 2.8023 5.4710 5.5419 1.2963 4.0 2.9620 3.0496 2.9579 5.4620 5.5402 1.4324 2.9724 3.0600 2.9462 5.4674 5.5456 1.4306 5.0 2.9599 3.0500 3.0440 5.4599 5.5425 1.5128 2.9703 3.0604 3.0323 5.4653 5.5479 1.5111

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

2.0 3.0489 3.1320 2.7233 5.5489 5.6195 1.2711 3.0717 3.1546 2.6995 5.5608 5.6313 1.2673 3.0 2.9656 3.0491 2.8139 5.4656 5.5366 1.2981 2.9760 3.0594 2.8023 5.4710 5.5419 1.2963 4.0 2.9239 3.0076 2.8611 5.4239 5.4951 1.3119 2.9299 3.0135 2.8543 5.4270 5.4981 1.3109 5.0 2.8989 2.9827 2.8901 5.3989 5.4702 1.3203 2.9028 2.9866 2.8856 5.4009 5.4722 1.3197

Table 5.6. Comparison of exact W

and approximate W

_q^*

for the T policy M/E

(E

,D)/1 and M/E

(E

,M)/1 queues with the second moment of the queue length.

M/E₃(E₄,D)/1 M/E₃(E₄,M)/1

T

=10

T

=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 2.8405 2.3560 17.0567 5.3405 4.3009 19.4659 2.8509 2.3682 16.9332 5.3459 4.3075 19.4248 0.4 3.1336 3.0319 3.2443 5.6336 5.2024 7.6526 3.1440 3.0438 3.1873 5.6389 5.2088 7.6276 0.6 3.7321 3.8012 1.8513 6.2321 6.0013 3.7037 3.7425 3.8127 1.8739 6.2375 6.0076 3.6861 0.8 5.6347 5.8102 3.1148 8.1347 7.8926 2.9768 5.6451 5.8205 3.1056 8.1401 7.8984 2.9688

μ Case 2. (λ=⁰^.³^, α=⁰^.⁰⁵^, β=³^.⁰^, γ =³^.⁰)

0.5 4.7922 5.2977 10.5476 7.2922 7.3724 1.0997 4.8027 5.3082 10.5272 7.2976 7.3783 1.1059 1.0 2.9656 2.7140 8.4845 5.4656 4.8120 11.9587 2.9760 2.7260 8.4012 5.4710 4.8184 11.9276 1.5 2.7829 2.3371 16.0190 5.2829 4.4185 16.3620 2.7933 2.3494 15.8916 5.2882 4.4250 16.3238

2.0 2.7283 2.1953 19.5346 5.2283 4.2662 18.4013 2.7387 2.2077 19.3871 5.2336 4.2727 18.3601 α ^{Case 3.} (λ=0.3, μ=1.0, β=3.0, γ =3.0)

0.05 2.9656 2.7140 8.4845 5.4656 4.8120 11.9587 2.9760 2.7260 8.4012 5.4710 4.8184 11.9276 0.10 2.9792 2.7165 8.8176 5.4792 4.8038 12.3261 2.9896 2.7284 8.7361 5.4846 4.8102 12.2954 0.15 2.9931 2.7191 9.1551 5.4931 4.7957 12.6955 3.0035 2.7310 9.0755 5.4985 4.8021 12.6652

0.20 3.0074 2.7218 9.4970 5.5074 4.7878 13.0669 3.0179 2.7336 9.4192 5.5128 4.7941 13.0371 β Case 4. (λ=0.3, μ=1.0, α=0.05, γ =3.0)

2.0 2.9735 2.2467 24.4404 5.4735 4.0112 26.7149 2.9839 2.2571 24.3578 5.4789 4.0167 26.6864 3.0 2.9656 2.2441 24.3293 5.4656 4.0127 26.5831 2.9760 2.2545 24.2458 5.4710 4.0182 26.5543 4.0 2.9620 2.2431 24.2714 5.4620 4.0137 26.5160 2.9724 2.2535 24.1874 5.4674 4.0192 26.4871 5.0 2.9599 2.2426 24.2358 5.4599 4.0144 26.4753 2.9703 2.2530 24.1515 5.4653 4.0199 26.4464

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

2.0 3.0489 2.7798 8.8285 5.5489 4.8857 11.9525 3.0717 2.8062 8.6426 5.5608 4.9000 11.8837 3.0 2.9656 2.7140 8.4845 5.4656 4.8120 11.9587 2.9760 2.7260 8.4012 5.4710 4.8184 11.9276 4.0 2.9239 2.6818 8.2815 5.4239 4.7754 11.9572 2.9299 2.6886 8.2346 5.4270 4.7790 11.9396 5.0 2.8989 2.6627 8.1487 5.3989 4.7535 11.9548 2.9028 2.6671 8.1187 5.4009 4.7558 11.9435

Chapter 6 Conclusions

In this thesis, we consider the policy and the policy for M/G/1 queues with server breakdowns and general distributed startup times, respectively. We develop the theoretical results for various system performance measures, such as the expected number of customers in the system, the expected length of the turned-off, complete startup, busy, and break-down periods, and the expected length of the busy cycle. In the

policy or policy M/G/1 queue with general service times and startup times, we

prove that the probability that the server is busy in the steady-state is equal to the traffic intensity

N T

ρ

. We construct a cost model to determine the optimal threshold or

T

so as to minimize this cost function. We also provide sensitivity analysis to discuss how the system performance measures can be affected by the changes of the input parameters (or cost parameters) in the investigated queueing service model. The sensitivity investigation is particularly valuable when evaluating future condition for the system analyst

N

We have utilized the maximum entropy principle to develop the maximum entropy (approximate) solutions for the policy M/G/1 queue with general repair times and startup times. We perform a comparative analysis between the approximate results obtained using maximum entropy principle and established exact results. We have demonstrated that the relative error percentages are very small (below 6.8%). For the policy M/G/1 queue with server breakdowns and general startup times, we have utilized the maximum entropy principle to develop the two maximum entropy (approximate) solutions for the policy M/G/1 queue with general repair times and startup times.

We perform a comparative analysis between two approximate results obtained using maximum entropy principle with different constraints and exact results. We have demonstrated that the relative error percentages are very small for maximum entropy solutions with the first moment of queue length (below 7.3%) and are very large for

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 64-0)