ANALYSIS OF M/M/C QUEUE MODEL UNDER N-POLICYKumar, Jitendra

ANALYSIS OF M/M/C QUEUE MODEL UNDER N-POLICY

Kumar, Jitendra

¹

, Sinde Vikas

²

and Kumar, Avanish

³

1

Department of Applied Mathematics, Madhav Institute of Technology & Science, GWALIOR -474005, M.P. (India)

[email protected]

2

Department of Applied Mathematics, Madhav Institute of Technology & Science, GWALIOR -474005, M.P. (India)

[email protected]

3

Department of Mathematical Sciences & Computer Applications, Bundelkhand University, JHANSI -284128, UP (India)

[email protected] ABSTRACT

Queue theory is an important and interesting field for the researchers, as it has many real life applications in production and inventory management. The Poisson input queue under N-policy and with a general startup and busy time for M/M/C need to be analyzed.

Here we have consider M/M/C queue model where customer's arrival rate varies according to the system status, which falls into one of the two cases that is start up and busy periods. In this paper the behavior of such as system under general conditions is studied with a view to class of arising in practical applications for the M/M/C queue model.

Index Terms- M/M/C queue model, N-Policy

INTRODUCTION

The M/M/C queue model under N-policy with a general setup time where the customer's arrival varies according to the system status, which falls into one of the two cases either set up or busy period. The server starts his setup immediately after the number of waiting customers reaches N.

Now the arrival based on poison process

with rate . The length of the set up time is generally distributed and independent of other random variables involved and is called set up period.

The purpose of this paper is to analyze

the optimal N-policy to minimize the system

cost in the M/M/C queuing system with gene-

rating function; here we use the multi-channel

queuing. Theory treats the condition in which

there are served service station in parallel and

each element in the waiting time can be served by more than one station. Each service facility is prepared to deliver the same type of service.

A single line usually breaks down into shorter lines in front in of each service station. The arrival rate  andservice rate  are mean values from Poisson distribution and exponential distribution respectively service discipline is FSFC, and contains are taken from a single queue, that is, any empty channel is filled by the next customer in line.

Objective :

Consider the Poisson Input Queue under N-Policy and with a general start up and busy time for M/M/C, on applying the gene- rating function with boundary value condition to obtain the set up time and busy period.

Mathematical Model and Notations :

n = Number of customers in the system, P

n

= Probability of n customers in the system, C = Number of parallel service channel,

 = Arrival rate of customer, R(z)= generation function, Q(z)= the service busy period,

µ = Service rate of individual channel, By queue length we mean only those in the queue, excluding the one being, if any,

  





0

dx x r

_n

P (z) = 



 1

0 N

n

P

n

_Z

ⁿ

 ^{x z} 

R , = 



 N n

n n

Z r

  z

R =   



0

, z dx x R

D*(s)  LST of service time (U) and startup time (V) respectively, we assume that the system is in steady state. The balance equation can be written as follows,

  ^t

P

_{n 1}

= ^C  ^P

n

  ^t n = 0 (1)

  ^t

dt R d

n

=     C   P

_n

  P

_n_c

 C  P

_n1

1



n ⁽²⁾

The boundary condition is

  ⁰

1 n

n

R

P

_



 (3)

Multiplying both side in (2) by equation by Z

ⁿ

, ,

2 ,

1 



 N N

n ---, we get.

 

ⁿ

n

t Z dt R

d =

 

_n ⁿ

n n

n c n n n

nZ R Z C R Z

R

C ₁

1 1

1

















 



^{ }



    

     

R



x z



C z z x R Z z x R

C _n

,

^c

, 

_n

,





   



 

 R x z 

dt

d , ₌

    R  x z 

C z Z

C

^c



_n

,





   

 (4)

Equation (4) can be solved by using the

boundary condition (3) which yield

 z  P

_n

Z

ⁿ

R 0 ,  

_1

1

0

c P

P 

 

₀

c P

₁

P 

 

2

1

c P

P 

  ^P

¹

^{c P}

²



  

2 1

0

P P

P    

 

  ce c sin

Now we use the method of variable and separation equation (4) can be written as

dt

d

) , (

) , (

z x R

z x

R =  ^{ }  ^{1  z}

^c

 ^{ c }  ^{1  z}

^1

 

^Log  ^R  ^{x , z}   ⁼

   

 ^{z x c z x} 

c 1

^c

1

¹

log       

^

   

C z x R Log ,

=

   

 ^{z x c z x} 

C exp .   1 

^c

  1 

^1

⁽⁵⁾

Now equation (5) x = 0, we get.

R(0, z) = C

So that R(0,z)=P

N-1

Z

ⁿ

=C

This value put in equation (5), we get.

 x z 

R , =

   

 ^{z x c z x} 

Z

P

_n1 ⁿ

exp .   1 

^c

  1 

^1

 (6)

Integrating (6), we get.

  





0

. ) , ( , z R x z dx x

R

= P

N-1

Z

ⁿ

   











 









 ^

dx e

e

^{zc x c z x}

0

1

1 ¹

.

^



P

N-1

Z =

ⁿ

   

  ^{  }

 

 

 



^{ }



^{ }







 ^

1 1 0

1 1

1

.

¹

dx e d x

z c

e

e

z x

x z c x z

c c









 

  ^ _

 





 







 ^

0

1 1

* 1

1

x dx z c

e

^{c z x}





=P

N-1

Z

ⁿ

   











  









 ^

dx e

e

^{zc x c z x}

0

1

1 ¹

.

1

^{ }

=P

N-1

Z

ⁿ

 

 ^z  ^{D  z }

c

z

^c



 

 



 

_1 ^*

1 1 1

Where ^{D *}  ^{   z}  ^  ^e

^

^

^1zc

^{ dt Since}

^{c}

     

 



 



 







_

1

* 1 1

1

1

z z D

z z z

P z

R



_n ⁿ



^c

 

(7) It is that R (z) is the probability generating function of the system size, when the system is as under setup and busy period.

Here the probability generating function of the

system size when the server is busy and P

0

is

the probability that the system is empty, that is

number of customer in the system.

Note:

) , ( x z x R





=   ^R  ^{x z} 

z z c

c

^c



_n

,





 

     





 (8)

Solving the partial differential equation (4)

R(x,z)= R   z e ^{ }

^c

^

^{zc cz}

^

^x

 



  

,



0

Since R  0 , z   C = ^P

n

^Z

ⁿ

  ^z

^c

 ^c  ^z   ^x

1

1

exp . 1 1

^



     



R(0,z)=P

N-1

Z

ⁿ

   ^r   ^x   ^{P x z} 

x z x

R , ,





 

  ₍₉₎

Equation (1) solving by partial differential

 ^{x z} 

P , =   ^{  }





x

du u r x

e z P 0 ,

⁰



Since R  0 , z   C   P

_n1

z

^N

The service busy period :

While supplementary variable system (technique) could be used. In this, start age, we prefer to consider a more elegant probabilistic approach to find. The Generating function R (z) of q

_n

.

Before going to find it let us find P [(B)], the probability that the system is in busy state

engaged in offering actual service. Consider that the service is on "extended vacation" during erodes.

Then

E [U] = N +  u =





The interval T

e

is equivalent to the length of the busy period of the standard M/G/1 queue generated by the customers arriving during (U). By conditioning on the number y of arrivals during (V), we find that

  _ ^





 





 



 

 

y E T E T

E

_e ^e

₌  ^ 



 

 u 1 N

=  ^ 



 1 

The instant of commencement of the busy period is a regenerative point. Using the well known result for the alternating renewal processes, we get

Pr [system in {B}] =  

   

_e

e

T E V E

T E



=

 

  



















 1 1

= 



 

That is Pr [{B}] =  (10)

Thus the fraction of time the server is

busy in offering actual service is , independent

of the control parameter N.

We shall now find Q (z). We have q

n

= Pr [system in {B}, queue length is n]

= Pr [queue length is n, in system {B}

* Pr (system in {B})]

Where dx = Pr (a departing customers leaves n behind)

Since PASTA holds for a Poisson input queue (from given reference [1])

  ^{z }  ^q

ⁿ

^z

ⁿ

Q

= ^  dx z

ⁿ

^  ^dxz

ⁿ

= ^   ^{z (11)}

where ^   z ^  dxz

ⁿ

REFERENCES

J. Medhi and J. G. C. Templeton

¹

, studies an M/G/1 queue under control operating policy and with a general startup time. It therefore considered necessary to examine an M/G/1 queue under N - policy and with a general start up time. We study such a system n this paper, Heyman

²

, ccording this paper an M/G/1 queue under a "control operating" (cop), here the server remains idle, after a system becomes empty, till the queue length builds up to reassigned desired level N (); this is known as N-policy, which has been shown by Heyman

²

to process certain optimal properties. Baker

³

considers an M/M/1 queue under N- policy and with start - up time, which the system requires before it becomes operation to do actual service. We

considered exponential start up time and obtained the steady state distribution of the queue length and observed that the addition at start up times renders the M/G/1 queue quite difficult to handle. Borthakur

^4,7

and

⁸

considered an M/M/1 queue under N-policy and with general start up time, they indicated same actual situation in queuing and inventory analyses where in such models could be useful.

Miller

⁶

, an M/G/1 finite queue on the busy period.

CONCLUSION

The model discussed in this paper is based on M/M/C queen model under N-policy with a general Start-up and busy time has been obtained. Here we have considered M/M/C queue model where customer's arrival rate varies according to the system status which fall into one of the two cases that is start up busy periods.

Here e obtained the results, Q ( z )   ( z ) &

     

 



 



 







_

1

* 1 1

1

1

z z D

z z z

P z

R 

_n ⁿ



^c

 

the problem discussed here has many applications in production and inventory.

REFERENCES

1. Medhi, J. and Templelotten, G.C., "A Poisson input queue under N-Policy and with a General startup time", Computers Opns, Res.,Vol. 19, No.-1, pp 35-41 (1992).

2. D.P. Heyman, "Optimum Operating Policies for M/G/1 Queuing systems", Ops.Res. vol.

16 pp 362-282 (1968).

3. K. P. Baker, "A note on operating policies for M/M/1 with exponential startups", Infor.

Vol. 11, pp 71-72 (1973).

4. A. Borthakar, J. Medhi and R. Gohain,

"Poisson Input Queuing Systems with Startup time and under Control Operating Policy", Computer OPS. RES. Vol. 14, PP 33-40, (1987).

5. J. Medhi, tochastic Models in Queuing theory, Academic Press, Boston, Mass.

6. E. Miller, "A note on the busy period of an M/G/1 Finite Queue", Ops. Res. Vol. 23, pp 1179-82 (1975).

7. A. Borthakar and R. Gohain, "On a Non-

Markovian queuing problem under a control operating policy and startup times", Aplik.

Mat. 27, pp 243-250 (1982).

8. A. Borthakar and R. Gohain,"Control policy and startup time for the multichannel queuing system". Paper presented in the third Anunal Conference of ISTPA (1981).

9. Chaudhry, M. L. and Templeton J. C. G.,

"A First Course in Bulk Queue", Jhon Wiley

and Sons, Inc. New York (1968).