Busy Period Analysis of M/M/2/∞ Interdependent Queueing Model With Controllable Arrival Rates And Feedback

International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: [email protected]

Volume 5, Issue 1, January 2016 ISSN 2319 - 4847

Volume 5, Issue 1, January 2016 Page 117

ABSTRACT In this paper the M/M/2/ queueing model with controllable arrival rates and feedback is considered. Distribution of busy period and mean length of busy period carried out for this model. The analytical results are numerically illustrated and the effect of the nodal parameters on the expected length of busy period is studied and relevant conclusions are presented. Keywords: Two server Markovian queueing model, Controllable arrival rates, Feedback, infinite capacity, Busy period Analysis. 1. Introduction In the earlier work, Thiagarajan and Srinivasan [6] have analysed the busy period analysis of interdependent queueing model with controllable arrival rates are employed and obtained the average length of the busy period for the model, Takagi and Tarabia [4] have studied a explicit probability density function for the length of a busy period for a finite capacity. Artalejo and Lopez Herrero [1] have analysed the busy period of the retrial queue with general service time distribution. Gopalan [2] has analysed the busy period analysis of a two-stage multi-product system. Soren Asmussen [7] has studied a busy period analysis, rare Events and Transient behaviour in fluid flow Models. Thangaraj and Vanitha [5] have analysed on the analysis of M/M/1 feedback queue with catastrophe using continued fractions. In this Chapter, busy period analysis of two server M/M/2/ interdependent queueing model with controllable arrival rates and feedback is considered. In section 2, the description of the queueing model is given stating the relevant postulates. In section 3, Differential-Difference Equations are derived. In section 4, Busy period analysis of the model and average length of the busy period of the model is obtained. In section 5, the analytical results are numerically illustrated and relevant conclusion is presented based upon a hypothetical data. 2. Description of the Model Consider two server infinite capacity queueing system with controllable arrival rates and feedback. Customer arrive at the service station one by one according to a bivariate Poisson stream with arrival rates (0-), (1-) (>0). There are two servers which provides service to all the arriving customers. Service times are identically and independently distributed exponential random variables with mean rate (-) (>0). After the completion of each service, the customers can either join at the end of the queue with probability p or customers can leave the system with probability q, p+q=1. The customer both newly arrived and those who opted for feedback are served in the order in which they join the tail of the original queue. It is assumed that there is no difference between regular arrivals and feedback arrivals. The customers are served according to the first come first served rule with a busy period as the interval of time from the instant customers arrive either with feedback or without feedback at an empty system and their service begins to the instant when the server becomes free for the first time. It is assumed that, The arrival process {X1(t)} and the service process {X2(t)} of the system are correlated and follow a bivariate Poisson distribution is given by

1 21 2i

m in x ,xx j x jjt

ij 0

1 1 2 21 2

e t t tP X t x , X t x

j! x j ! x j !

Busy Period Analysis of M/M/2/ Interdependent Queueing Model With

Controllable Arrival Rates And Feedback

G. Rani1, A. Srinivasan2

1Associate Professor, PG and Research Department of Mathematics, Seethalakshmi Ramaswami College, Tiruchirappalli, 620002. Tamil Nadu, India.

2Associate Professor, PG and Research Department of Mathematics,

Bishop Heber College (Autonomous), Tiruchirappalli, 620017. Tamil Nadu, India.




where x1, x2 = 1, 2,..., and 0, 1, >0, 0 < min (i,), (i=0,1) with parameters 0, 1, and for mean faster rate of arrivals, mean slower rate of arrivals either with feedback or without feedback, identical mean service rate for first server, second server and mean dependence rate respectively Also the state space of the system is

s = {1, 2, 3,..., r–1, r, r+1, ..., R–1, R, R+1,...,k–1, k, k+1,...,}

Postulates of the model are

1. Probability that there is no arrival and no service completion during a small interval of time h, when the system is in faster rate of arrivals either with feedback or without feedback, is

01 2p 2q h o h

2. Probability that there is one arrival and no service completion during a small interval of time h, when the system is in faster rate of arrivals either with feedback or without feedback, is

(0 – )h + o(h) 3. Probability that there is no arrival and no service completion during a small interval of time h, when the

system is in slower rate of arrivals either with feedback or without feedback is,

11 2p 2q h o h

4. Probability that there is one arrival and no service completion during a small interval of time h, when the system is in slower rate of arrivals either with feedback or without feedback is

(1 – ) h + o(h) 5. Probability that there is no arrival and one service completion during a small interval of time h, when the

system is either in faster or is in slower rate of arrivals either with feedback or without feedback is

2p 2q h o h

6. Probability that there is one arrival and one service completion during a small interval of time h, when the system is either in faster or slower rate of arrivals either with feedback or without feedback is

0 1 2p 2q h o h

3.Differential - Difference Equations

Pn(t0): the probability that there are n customers in the system at time t0 when the system is in faster rate of arrivals either with feedback or without feedback.

Pn(t1): the probability that there are n customers in the system at time t1 when the system is in slower rate of arrivals either with feedback or without feedback.

we observe that Pn(t0) exists when n = 0, 1, 2, 3,...,r–1; both Pn(t0) and Pn(t1) exist when n = r+1, r+2, r+3, ..., R–2, R–1; only Pn(t1) exists when n = R, R+1, R+2, ..., Assume that the initial system size when the system is in faster rate of arrivals is 1 either with feedback or without feedback and that of when the system is in slower rate of arrivals is r+1.

Let 0 0P t' be the busy period density when the system is in faster rate of arrivals either with feedback or

without feedback and r 1 1P t' be the busy period density when the system is in slower rate of arrivals either with

feedback or without feedback. The differential-difference equations governing the system size with an absorbing barrier imposed at zero

system size when the system is in faster rate of arrivals either with feedback or without feedback and with an obsorbing barrier imposed at (r+1) system size when the system is in slower rate of arrivals either with feedback or without feedback, are

0 0 1 0P t 2 P t '

(because of the absorbing barrier) ... (3.1)

1 0 0 1 0 2 0P t 2q P t 2q P t '


n 0 0 n 0 n 1 0P t 2q P t 2q P t '

0 n 1 0P t n = 2,3,4,...,r–1 ... (3.3)




r 0 0 r 0 0 r 1 0P t 2q P t P t '

r 1 0 r 1 12q P t 2q P t ... (3.4)

n 0 0 n 0 0 n 1 0P t 2q P t P t '

n 1 02q P t n = r+1, r+2,..,R–2 ... (3.5)

R 1 0 0 R 1 0 0 R 2 0P t 2q P t P t '


r 1 1 r 2 1P t 2q P t '


r 2 1 r 2 1 r 3 1P 1 2q P t 2q P t '


n 1 1 n 1 1 n 1 1P t 2q P t P t '

n 1 12q P t n = r+3, r+4,..., R–1 ... (3.9)

R 1 1 R 1 0 R 1 0P t 2q P t P t '

1 R 1 1 R 1 1P t 2q P t ... (3.10)

n 1 1 n 1 1 n 1 1P t 2q P t P t '

n 1 12q P t n = R+1, R+2, R+3,... ... (3.11)

Let

00

2 2q

and

11

2 2q

where 02

is faster rate of arrivals intensity and

12

is slower rate of arrivals intensity.

4. Busy Period Analysis of the Model

Define R 1

n0 n 0

n 0P z, t P t z

... (4.1)

n1 n 1

n r 1P z, t P t z

... (4.2)

Such that the summation is convergent in and on unit circle. When (3.1) to (3.6) are multiplied through by zn

for n = 0, 1, 2,...,R–1 and summing over from n = 0, 1, 2,..., R–1, it is found that

R 1

' n0 n 0

n 00

d P z, t P t zdt

R 1 R 1 R 1

n0 n 0 0 n 1 0 n 1

n 0 n 0 n 02q P t P t 2q P 0 z

+ r 1 r 1

r 1 1 r 1 02q P t z 2q P t z

0 0 02q z 2q P 0z

R 1 R 1

n n 10 n 0 0 n 1 0

n 0 n 12q P t z z P t z




R 1n 1 r 1

n 1 0 r 1 1n 0

2qP t z 2q P t z

z

0 0 0 0zz 2q z 2q P t

z

20 0 0 0

0

dz P z, t z 2q z 2q P z, tdt

20 0 0 0z 2q z 2q P t

r 1r 1 12q z P t ... (4.3)

Taking Laplace Transform on both sides of (4.3) we get

20 0 0 0 0z s P z,s P z,0 z 2q z 2q P z,s

20 0z 2q z

r 1r 10 10

2q P s 2q z P s

2 r 10 r 10 0 1

0 20 0 0

z 1 z 2q z P s 2q z P sP z,s

2q s z 2q z

where

0 0L P z, t P z,s ... (4.4)

0 r 10 0 0 r 1 1 1L P t P s ,L P t P s and P z,0 z

Since the Laplace Transform 0P z,s converges in the region z 1, Real 0s 0, wherever the

denominator of the right hand side of (4.4) has zeros in that region, so must the numerator, the denominator has two zeros, that is

20 0 0z 2q s z 2q 0

20 0 0 0 00

10

2q s 2q s 4 2qz

2

20 0 0 0 00

10

2q s 2q s 8 qz

2

20 0 0 0 00

20

2q s 2q s 4 2qz

2

20 0 0 0 00

20

2q s 2q s 8 qz

2

and hence

20 0 0 0 00

10

20 0 0 0 00

20

2 q s 2 q s 8 qz

2

2 q s 2 q s 8 qz

2

... (4.5)




From (4.5), It is clear that 0 01 2z z

0 0 0 0 0 01 2

0 0

2 4q 2s 2q sz z

2

0 01 2

0

2qz z

and hence

0 0 0 01 2

0

0 01 2

0

2q sz z

2qz z

... (4.6) Using Rouche’s theorem in (4.4) we get

2 r 10 0r 11 1 1

0 0 00 1

z 2q z P sP s

s z

... (4.7)

Using the result (4.7) in (4.4) we get

2 r 10 0r 11 1 12

0 00 1

r 1r 1 1

0 0 00 2 1

z 2q z P sz 1 z 2q z

s z

2q z P sP z,s

z z z z

...

(4.8)

2 r 10 0r 11 1 1

00 1

0 0 0 0 00 1 2

z 2q z P s2q

s zP s P 0,s

z z

=

r

r

r 1 10 000 0 2 0 2

2q 2q 2qP s

s z s z

r

r

00

r 1 10 00 0 2 2

2q2q 1 P ss z z

r

rr 1 1

00 0 0 000 2 2

2q 2q P ss P s 2q

z z

... (4.9)

From equation (4.9) we get

r

rr 1 1

00 0 0 00 0 00 2 2

2q 2q P sd ds P s 2qds ds z z

0

00 0 s 00 0 0

2qd s P sds 2q




0

r

r 1 10 0s 0

2q r2q P s2q

0

r

r 1 1s 00 0 0

2q 2q1 .r Lt P s2q

0 0

r

0 r 10 0 1s 0 s 00 0 0 0

2q 2qd 1s P s r Lt P sds 2q

... (4.10)

0

'00 0 0 0 0 0s 0

0 0

d ds P s L P t sds ds

0 0

0

s t '0 0 0

0 0 S 0

d e P t dtds

0 0

0

s t '0 0 0 0

0 S 0

e t P t dt

0'0 0 0 0 busy

0

t P t dt E T

0 0

000 0 busyS

0

d s P s E Tds

... (4.11)

From equations (4.10) and (4.11) we get

0

r0

r 1busy 1s 00

1 1 2E T .r. Lt P s0 01

2

The average length of the busy period when the system is in faster rate of arrivals either with feedback or without feedback is given by

0

r0

r 1busy 1s 00

1 1 2E T .r. Lt P s0 01

2

... (4.12)

which is consistent with the corresponding result of the conventional model discussed by Gross and Harris (1974) for the case = 0, p=q=1, 0 = 1 = and is also consistent with the corresponding result of the busy period analysis of M/M/1/ model discussed by Thiagarajan and srinivasan for the case p=q=1. From (3.6) to (3.10) are multiplied through by zn, n = r+1, r+2, r+3,..., R–1, R, R+1,..., and summing on n from n = r+1, r+2,...R–1, R+1,..., it is found that

n1 n 1

n r 11

d P z, t P t zdt

' ... (4.13)

Using the procedure as in the case when the system is in faster rate of arrivals either with feedback or without feedback we get

r 1 r 2 R 11 r 1 1 r 2 1 R 1 1

1

d P z, t P t z P t z ... P t zdt

' ' '

R R 1R 1 R 1 1P t z P t z ...

' '




n1 1 n 1

n r 1

2q2q z P t z

z

r 11 1 r 1 1

2q2q z P t z

z

R0 R 1 0P t z

1 1 1

2q2q z P z, t

z

r 11 1 r 1 1

2q2q z P t z

z

R0 R 1P 0 z

2 n1 1 1 n 1

n r 11

dz p z, t z 2q z 2q P t zdt

2 r 11 1 r 12q z 2q z P 1 z

R 10 R 1 0P t z

Taking Laplace Transform on both side we get

21 1 1 1 1z s P z,s P z, r 1 z 2q z 2q P z,s

21 1z 2q z

r 1 R 1r 1 R 11 0 02q P s z P s z

r 2 2 r 1r 11 1 1

R 1R 10 0

z z 2q z 2q P s z

P s z

21 1 1 1z 2q s z 2q P z,s

r 2 r 1 R 1r 1 R 11 1 0 0

1 21 1 1

z 1 z 2q z P s P s zP z,s

2q s z z 2q

... (4.14) where

r 1r 11 1 r 1 1 1L P z, t P z,s ,L P t P s and P z, r 1 z

Since the Laplace Transform 1P z,s Converges in the region z 1, Real (s1)0, wherever the

denominator of the right hand side of (4.14) has zeros in that region, so must the numerator, the denominator has two zeros. Using Rouehe’s theorem in (4.14) we get

r 1 R1 1

R 11 0 0 1r 1 1

1

z P s zP s

s

... (4.15)

r 2 r 1 R 1r 11 1 0 R 1 0

1 1 11 1 2

z 1 z 2q z z P s z P sP z,s

z z z z

... (4.16)




where

21 1 1 1 11

11

21 1 1 1 11

21

2q s 2q s 8 qz

2

2q s 2q s 8 qz

2

... (4.17)

R 11 1 R 1 0 0L P z, t P z,s ,L P t P s

It is clear that 1 11 2z z , from equation (4.17)

1 1 1 11 2

1

1 11 2

1

2q sz z

2qz z

... (4.18)

Using (4.15) in (4.16) we get r 2 r 1 R 11 1 1 1 1

r 1 R 11 1 1 1 1 1 0 0 1z 1 z 2q z z P s P s z 0

r 21 R 1R 11 0 0 1

r 1 1 1 1 r 11 1 1 1

z P s zP s

1 z 2q z z

r 11 R 1R 11 0 0 1

r 1 1 1 1 r1 1 1

z P s zP s

1 z 2q z z

... (4.19)

From equations (4.15), (4.17), (4.18) and (4.19) we get

r 1 R 1

R 1

1 1 r 1R 11 0 0 1r 2

1 1 1 r1 1 1 1

R 10 01 1 1

1 1 2

z P s z zz 1 z 2q z

1 z 2q z z

z P sP z,s

z z z z

r 1 R 11 1 r 1R 11 0 0 1 1r 2

1 r 11 1

R 1R 10 0

1 1 11 1 2

z P s z zz 1 z 2q z

s z

z P sP z,s

z z z z

r 1 R1 1R 11 0 0 1r 2

11

R 1R 10 0

1 1 11 1 2

z P s zz 1 z 2q z

s

z P sP z,s

z z z z




r 1 R1 11 0 R 1 0 1

1 1 11 1 2 1

z P s z2qP 0,s

z z s

r 1 R1 1

R 11 0 0 11

1

z P s zP 0,s

s

... (4.20)

r 1 R1 1R 11 1 1 0 0 1s P 0,s z P s z

Rr 1

R 10 01 112 12

2q 2q1 P sz z

R

r 1 R

R 11 1 0 01 11 12 2

2q 2q1 1s P 0,s P sz z

r 1

1 11 1 1

2qd 1s P 0,s r 1ds 2q

1

R

R 10 0S 01

2qLt P s R

1

r 1

1 1 S 01 1 1

2qd 1s P 0,s r 1ds 2q

1

R

R 10 0S 01

2qLt P s R

... (4.21)

1 1

11

S t '11 1 r 1 1 1S 0

t 1 0 S 0

d ds P s e P t dtds ds

1 1

1

S t1 r 1 1 1

0 S 0

e t P t dt

11 r 1 1 busy

0

t d P t E T

111 1 busy

1

d s P s E Tds

... (4.22)

From (4.21) and (4.22) we get

r 11

busy1 1

2q1E T r 12q

1

R

R 10 0S 01

2qLt P s R

and hence




The average length of the busy period when the system is in the slower rate of arrivals either with feedback or without feedback, is

1

r 1 R1

R 1busy 0 0S 0

1 2 2E T r 1 Lt P s .R1 12q 1 1

... (4.23) which is consistent with the corresponding result of the conventional model discussed by Gross and Harries (1974) for case = 0, 0 =1 = and p=q=1, the present model reduces to busy period analysis of M/M/1/ interdependent model with controllable arrival rates discussed by Thiagarajan and Srinivasan [6] for the case p=q=1, Two server = single server (2q=q).

5. Numerical Illustrations For various values of r, R, 0, 1, and , 0

busyE T and 1busyE T are computed and tabulated by taking

p=q=1 .2

Table 5.1

R R 0 1 0busyE T 1

busyE T

5 10 5 4 9 0.5 1.888 164.10 5 10 5 4 9 0.0 1.800 103.79 5 10 5 4 9 1.0 2.000 287.67 5 10 5 4 11 0.5 1.555 416.57 5 10 5 4 11 0.0 1.460 247.15 5 10 5 4 11 0.5 1.666 937.29 5 10 8 4 11 0.5 1.866 416.57 5 10 5 4 11 1.0 1.666 783.85 6 10 8 4 12 0.5 1.916 305.08 6 14 8 5 12 0.5 1.916 913.86 6 14 8 5 12 1.0 2.054 1189.40 8 14 9 6 12 0.5 3.156 509.24 8 13 9 6 12 0.0 3.111 341.33 8 13 8 6 12 1.0 2.750 804.85

i. It is observed from the Table 5.1 that the average length of the busy period is increase when the system is either in

faster or in slower rate of arrivals by increasing the value of r while the other parameters were fixed. ii. The average length of the busy period rapidly increases when the system is in faster rate of arrivals by increasing

the value of 0 and keeping the other parameters were fixed. iii. The average length of busy period rapidly decreases by increasing the value of and keeping the other parameters

were fixed while the system is in faster rate of arrivals. iv. The average length of busy period increases by increasing the value of and keeping the other parameters were

fixed while the system is in slower rate of arrivals. v. The average length of busy period increases while the system is in faster rate of arrivals by increasing the value of

keeping the other parameters were fixed. vi. The average length of busy period increases while the system is in slower rate of arrivals by increasing the value of

and keeping the other parameters were fixed. vii. The average length of busy period decreases while the system is in slower rate arrival by decreasing the value of R

keeping the other parameters were fixed. viii. The average length of busy period is remain unchanged while the system is in faster rate of arrivals by increasing

the value of R keeping the other parameters were fixed. References

[1] Artalejo, J.R. and Lopez-Herrero, M.J. On the Busy Period of the M/G/1 Retrail Queue, Naval Res. Logist, 47, 115-127, (2000).




[2] Gopalan, M.N. On the busy period analysis of the two stage multi-product system, International Journal of Management and Systems, 11, 249-266. (1995).

[3] Srinivasan, A. and Thiagarajan, A Finite Capacity Multi Server Poisson Input Queue with Interdependent Inter-arrival and Service Times and Controllable Arrival Rates, Bulletin of Calcutta Mathematical Society, 99(2), 173-182, (2007).

[4] Takagi, H. and Tarabia, H. Explicit Probability Density Function for the Length of a Busy Period in an M/M/1/k Queue, Int. Advances in Queueing Theory and Network Applications, 213-226, (2009).

[5] Thangaraj, V. and Vanitha, S. On the Analysis of M/M/1 Feedback Queue with Catastrophes using continued Fractions, Int. Jr. of Pure and App. Math., 53(1), 133-151, (2009).

[6] Thiagarajan, M. and Srinivasan, A. Busy Period Analysis of M/M/1/ Interdependent Queueing Model with Controllable Arrival Rates, Acta Ciencia India, 33M(1), 19-24, (2007).

[7] Soren Asmussen,”Busy period analysis, rare events and transient behavior in Fluid Flow models”, Journal of Applied Mathematics and Stochastic Analysis,Vol.3,269-299,(1994).

AUTHOR

Dr.G.Rani is an associate professor in Seethalakshmi Ramasami College (Autonomous),Trichy.She received Ph.D. degree in Mathematics from Bharathidasan University. Her research focuses on Stochastic models in Queueing Theory with feedback.

Documents

Busy Period Analysis of M/M/2/∞ Interdependent Queueing Model With Controllable Arrival Rates And Feedback