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

8/10/2019 ANALYSIS OF M/M/C QUEUE MODEL UNDER N-POLICY

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

[email protected] 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 arrivalrate varies according to the system status, which falls into one of the twocases 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 classof arising in practical applications for the M/M/C queue model.

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

J. Comp. & Math. Sci. Vol. 1(1), 27-32 (2009).

INTRODUCTION

The M/M/C queue model under N-policy with a general setup time where thecustomer's arrival varies according to the systemstatus, which falls into one of the two caseseither set up or busy period. The server startshis 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 isgenerally distributed and independent of other random variables involved and is called set up

period.

The purpose of this paper is to analyzethe optimal N-policy to minimize the systemcost in the M/M/C queuing system with gene-rating function; here we use the multi-channelqueuing. Theory treats the condition in whichthere are served service station in parallel and


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each element in the waiting time can be served by more than one station. Each service facilityis 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 arrivalrate and service rate are mean values from

Poisson distribution and exponential distributionrespectively service discipline is FSFC, and contains are taken from a single queue, that is,any empty channel is filled by the nextcustomer in line.

Objective :

Consider the Poisson Input Queueunder 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,

= Service rate of individual channel,

By queue length we mean only those in thequeue, excluding the one being, if any,

0

dx xr n

P ( z) =1

0

N

nnP

n Z

z x R , = N n

nn Z r

z R = 0

, dx z x R

D*(s) LST of service time (U) and startuptime (V) respectively, we assume that thesystem is in steady state. The balance equationcan be written as follows,

t Pn 1 = t PC n n = 0 (1)

t Rdt d

n = 1 ncnn PC PPC

1n (2 )The boundary condition is

01 nn RP (3)

Multiplying both side in (2) by equation by Z n,

,2,1 N N n -------, we get.

nn Z t Rdt d

=

nnnn

ncn

n

nn Z RC Z R Z RC 1

111

z x R z

C z x R Z z x RC nc

n ,,,

z x Rdt d

, =

z x R z

C Z C nc ,

(4)

Equation (4) can be solved by using the


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boundary condition (3) which yield

n z 1,

10 PcP 10 Pc

P

21 PcP 21 Pc

P

210 PPP

c

cesin

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

dt d

),(),(

z x R z x R = 111 zc z c

z x R Log , =

x zc x zc c 111log

C z x R Log ,

=

x zc x zC c 111.exp (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.

z x R , = x zc x z Z P cnn 11 11.exp (6)

Integrating (6), we get.

0

.),(, dx z x R z x R

= P N -1 Z n dxee x zc x zc

0

11 1.

=P N -1 Z n

101

11

1.

edxd

x zcee x z

x zc x zc

01

1

1* dx

x zce x zc

= P N -1 Z n dxee x zc x z

c

0

11 1.1

= P N -1 Z n z D zc z

*11

11

Where dt e z D c z1* Since

c

1

1*111

z z D z z zP z R cnn

(7 )

It is that R (z) is the probabilitygenerating function of the system size, whenthe system is as under setup and busy period.Here the probability generating function of thesystem size when the server is busy and P 0 isthe probability that the system is empty, that isnumber of customer in the system.


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Note:

),( z x R x

= z x R zc

zc nc ,

(8)

Solving the partial differential equation (4)

R( x,z)= x zc zc

e z R

,0

Since C z R ,0 = x zc z Z P cnn 11 11.exp

R(0, z)= P N-1 Zn

z xP xr x

z x,

, (9)

Equation (1) solving by partial differential

z xP , =

x

duur x

e zP 0,0

Since N n zPC z R 1,0

The service busy period :

While supplementary variable system(technique) could be used. In this, start age, we

prefer to consider a more elegant probabilisticapproach to find. The Generating functionR (z) of qn.

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" duringerodes.Then

E [U] = N + u =

The interval T e is equivalent to the length of the busy period of the standard M/G/1 queuegenerated by the customers arriving during(U). By conditioning on the number y of arrivalsduring (V), we find that

y

T E E T E ee =

1

u N

=

1

The instant of commencement of the busy period is a regenerative point. Using the wellknown result for the alternating renewal

processes, we get

Pr [system in {B}] = ee

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

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of the control parameter N.

We shall now find Q (z). We have

qn = 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 leavesn behind)

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

nn zq zQ =

n zdx ndxz

= z (11)where ndxz z

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 thereforeconsidered necessary to examine an M/G/1queue 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 queueunder a "control operating" (cop), here theserver 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 processcertain optimal properties. Baker 3 considers anM/M/1 queue under N- policy and with start - uptime, 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 thequeue length and observed that the addition atstart up times renders the M/G/1 queue quitedifficult 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 inventoryanalyses 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-policywith a general Start-up and busy time has been

obtained. Here we have considered M/M/Cqueue model where customer's arrival rate variesaccording to the system status which fall intoone of the two cases that is start up busy periods.Here e obtained the results, )()( z zQ &

11

*111 z z D z z zP z R cnn

the problem discussed here has manyapplications in production and inventory.

REFERENCES

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

2. D.P. Heyman, "Optimum Operating Policiesfor M/G/1 Queuing systems", Ops.Res. vol.16 pp 362-282 (1968).

3. K. P. Baker, "A note on operating policiesfor M/M/1 with exponential startups", Infor.Vol. 11, pp 71-72 (1973).

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4. A. Borthakar, J. Medhi and R. Gohain,"Poisson Input Queuing Systems with Startuptime 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 controloperating policy and startup times", Aplik.Mat. 27, pp 243-250 (1982).

8. A. Borthakar and R. Gohain,"Control policyand startup time for the multichannel queuingsystem". 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 Wileyand Sons, Inc. New York (1968).

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