M/G/1 Queueing System with Extended Vacation, Service Interruption ...

Journal of Computer and Mathematical Sciences, Vol.6(7),363-370, July 2015 (An International Research Journal), www.compmath-journal.org

ISSN 0976-5727 (Print) ISSN 2319-8133 (Online)

M/G/1 Queueing System with Extended Vacation, Service Interruption, Delay Time and Stages in Repair S. Maragathasundari and S. Sowmiyah Asst. Prof, Department of Mathematics, Velammal Institute of Technology, Chennai, INDIA. [email protected] (Received on: July 12, 2015) ABSTRACT

In this paper a / /1 Queueing model with service interruption, Repair process and delay in repair is described. On completion of a service, the server will go for a vacation .An additional aspect of Optional extended vacation is considered in this model .The system subject to breakdown at random. In this model, repair process does not start immediately. There is a delay in getting into the repair process. Repair takes place in two stages. Moreover service time, Vacation time, Delay time & Repair time follows general distribution. Steady state solution & Performance measures are derived. Mathematics Subject Classification: 60K25, 60K30 Keywords: Random breakdown, Repair process, Extended vacation, Steady state, Queue size distribution.

1. INTRODUCTION In recent years, several mathematicians became interested in Queueing systems with server vacations and/or random system breakdowns and developed general models which could be used in more complex situations. Numerous researchers have studied non Markovian queueing systems. The single arrival queueing systems M/G/1 have been studied by numerous researchers Madan et al.10 Madan and Choudhury12. Choudhury and Madan5, investigated a two phase arrival queueing system with a vacation time under Bernoulli schedule. Chodhury and Madan1, studied a two stage batch arrival queueing systems with a modified Bernoulli schedule vacation under N-policy. Choudhury9 have studied vacation queues. Anabosi & Madan6 studied a single server Queue with two types of service, Bernoulli schedule server vacations and a single vacation policy. In this paper we study a Non Markovian arrival queueing system / /1 in which, after every service completion June, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org

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S. Maragathasundari, et al., J. Comp. & Math. Sci. Vol.6(7), 363-370 (2015)

the server has the option to leave for a vacation. Moreover the server has a option of taking a extended vacation with probability r or continue service with probability 1-r. Moreover, we assume that the server may breakdown randomly. The repair process does not necessarily start immediately after a breakdown, thus there may be a delay before starting repairs. We assume that the service times, vacation times, extended vacation times, repair times and delay times each have a general distribution while the breakdown times are exponentially distributed. This paper is organized as follows. The model assumption is given in section 2. In section 3 all the equations governing the mathematical system in the steady state are formulated. The supplementary variable technique is used in this section to obtain the closed form of the probability generating function of the queue length. The average queue size and the average waiting time are given in section 4. Conclusions are given in section 5. 2. MODEL ASSUMPTIONS a) Customers arrive at the system in batches of variable size in a compound Poisson process and they are provided one by one service on a ‘first come’-first served basis. Let 1,2,3 … . be the first order probability that a batch of customers arrives at the system during a short interval of time , , where 0 1 and ∑ 1 and 0 is the mean arrival rate of batches. b) Each customer undergoes two stages of service provided by a single server on a first come first served basis. The service time follows general(arbitrary)distribution with distribution function and density function . Let be the conditional probability density of service completion during the interval , ,given that the elapsed time is ,so that

1 1 ! and therefore " # $) %&'& 1,2 2 c) As soon as a service is completed, the server may go for a vacation. Also the server may go for a extended vacation with probability r or return back to the system to continue the service for the next customer with probability 1-r. d)The server’s vacation time follows a general(arbitrary) distribution with distribution function and density function *. Let + be the conditional probability of a completion of a vacation during the interval , , so that (

+

* 1 !

* +" # $) ,&'&

And, therefore

(

July, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org

3

4


365

The server’s Extended vacation time follows a general(arbitrary) distribution with distribution function and density function .. Let / be the conditional probability of a completion of a Extended vacation during the interval , , so that . /

30 1 ! And, therefore . /" # $) 1&'&

40

(

e) The system may breakdown at random and breakdowns are assumed to occur according to a Poisson stream with mean breakdown rate 2 0. f) The server’s delay time follows a general(arbitrary) distribution with distribution function 3 and density function 4. Let 5 be the conditional probability of a completion of a vacation during the interval , , so that Delay time 5

6&

#7&

and 4 5" # $) 8&'& (

g) The server’s repair time follows a general(arbitrary) distribution with distribution function 9 and density function : . Let ; be the conditional probability of a completion of a repair during the interval , , so that = Repair time ; #> &

< & =

1,2 and : ; " # $) ?= &'& (

(5)

h) Various stochastic process involved in the system are assumed to be independent of each other. 3. EQUATIONS GOVERNING THE SYSTEM The equations governing the system are as follows: @ @ AB 2 AB AB#

@& @

@& @

A D 2 EAC 0 @& C @& @ @ FB + FB FB# @& @& @ F + FG 0 @& G @ @ H / HB HB# @& B @& @ @ H @& / HC 0 @& C @ I ; IB IB# @& B @ J J J I ; IB IB# @& B @ I ; IC 0 @& C @ J J I ; IC 0 @& C

(6) (7) (8) (9) (10)


(11) (12) (13) (14) (15)

366

@ K @& B @ K @& C


@ @&

0

5 KB KB#

(15a) (16)

L $C IC J ; $C AC $C FC + $C HC / The following boundary Conditions are used to solve the above equations:

AB 0 $C IBM ;J $C ABM 1 ! N $C FBM + $C HBM /

J

FB 0 $C AB HB 0 N $C FB + KB 0 2 $C AB# IB 0 $C KB 5 J IB 0 $C IB ;

(17)

KC 0 0 IC 0 0 Multiplying Equation (6) by O B sum over P from 1 Q ∞ and adding to (7), we obtain @ A , T ! T 2AS , T 0 @& S Similarly, @ FS , T D ! T + EFS , T 0 @& @ H , T D ! T / EHS , T 0 @& S @ K , T D ! T 5 EKS , T 0 @& S @ I , T D ! T ; EIS , T 0 @& S @ J J I , T D ! T ;J EIS , T 0 @& S

(18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31)

Multiplying Equation (18) by T BM , sum over P from 0 Q ∞, TAS 0, T $C IS , T;J 1 ! N $C FS , T+ $C HS , T/ (32) $C AS , T ! L FS 0, T $C AS , T HS 0, T N $C FS , T+

KS 0, T 2 $C AB# , T IS , T $C KS , T5


(33)

(34) (35) (36)


IS , T $C IS , T; Integrating Equation (26) from 0 to ∞

J

367 (37)

AS , T AS 0, T" #U#UVMW&#$) %X'X Again Integrating the above equation, by parts with respect to , Y

(38)

#[\]

AS T AS 0, T Z (39) _ ^ \ Where ` ! T 2 & is Laplace Stieltjes Transform of the service time. Multiply both sides of Equation (27) by & integrating over , we get (40) $C AS , T AS 0, T\ Carrying out the same process for Vacation, Extended vacation time & repair time and Delay time we have, b * (41) $C FS , T+ FS 0, T (42) $ HS , T/ HS 0, Tc * C $C IS

\\\ * , T; IS 0, T9

(43)

J J \\\J* (44) $C IS , T;J IS 0, T9 b (45) $C KS , T5 KS 0, T3 * Substituting the equations (40),(41),(42),(43) in eq (32) we get, 1 ! \ b *2T b *AS 0, T c *N b * \\\ *3 TAS 0, T 9 AS 0, T \ ` AS 0, T\ 1 ! NAS 0, T\ 46 c b \ b \ \ AS 0, T eO ! * ! N * * ! 1 ! Nf \\\ *3 b *2T1 ! \ !L !9 (47)

AS 0, T eh#ibj[\]#kibjlc

#U]g \ ]##k[\ ]f#> b jWV#[\] \\\\ j[ m j7

(48)

Let `S T be the probability generating function of the queue size J `S T AS T FS T HS T IS T IS T KS T

#Ug j

b jEM]ki b j[\m ]D#lc jE jD#[\m ]EM][\m ]D#i \\\\ \m ]#[\o ]EM MWVD#[\o ]E#WV> p m jD[

#Ugn

\\\\ \\\\ \o ]D#[\m ]E WV> m j> o j[ b j[\m ]#ki b jlc j[ \\\\ \\\\ ]DV#i m ]##k[ m ]E q r \\\\ \o ]WVD#[\m ]E #> m j[

49

b *E N b *\ D1 ! c *E *D1 ! \ E \ D1 ! \\\ *D\ ! \JE Where tT !L n 2TD1 ! \JE ! 2T9 p \\\ *9 \\\J *\J D1 ! \ E 2T9


368


\\\ ! 1 ! N \\\ E b *\ ! N b *c * DT ! KT * q r \\\ *\J 2TD1 ! \ E !9

Using the condition `S 1 L 1 the unknown factor L can be determined. Also the utilization factor u can be determined.

4. THE AVERAGE QUEUE SIZE AND THE AVERAGE WAITING TIME

Let vS denote the mean number of customers in the queue under the steady state. Then vS w `S Tx T V Since `S T =0⁄0 at T 1, we use the result vS lim AS T V} T K ~ Tt ~~ T ! t ~ TK ~~ T

AS 1 lim J V} 2DK ~ TE K ~ 1t ~~ 1 ! t ~ 1K~~ 1

50 J 2DK ~ 1E t ~ 1 !2H* ! 22H ! 22 (51) ~~ 1 J w \ 2 t

!Le!2H H ! 22H* \ 2H J !J 2H*J\ 2 ! 2N2\2H J N2 J J J \ ! 2H(3 ! 32H9 2H ! 22H ! 22 !22JH9 H ! 22H9 D\2 ! H3E !22J HH3 ! 22J H9J H ! 22J H( J (52) ~ 1 K

!e\2f (53) K ~~ 1 !e!wD1 ! \2 ! N\ 2 ! \2 N\2E \\\\~ *\2 ! b *\2f 1 ! e! \\\\~ *c*\2 ! \\\~ *\2 ! b * b *c *\2f+ 2 !Ne! \ !1 ! N!2 2D1 ! \ 2 ! \2E 2 ! 2H\2 22H 2H ! q r! !N2H ! N2H ! N J 2H J H9 2H J H32H 2 wH9 D1 ! HE 2H3D1 ! HE (54) w!2D1 ! HE ! 2D1 J HJ E

Where primes and double primes in (43) denote first and second derivatives at T 1, respectively.Carrying out the derivatives at T 1 and if we substitute the values of t ~ 1, t ~~ 1, K ~ 1 0PK ~~ 1 into (50) we obtain vS in closed form. Further, the mean waiting time of a customer could be found using 3S vS ⁄ . July, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org


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5. CONCLUSIONS

In this paper we have studied a //1 queue with extended vacation, service interruption, delay times, stages in repair. The probability generating function of the number of customers in the queue is found using the Supplementary variable method. This paper clearly analyses the steady state results and some performance measures of the queueing system. The result of this paper is useful for computer communication network, and large scale industrial production lines.

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