A single queue is serviced by two servers alternately, in alternate busy periods. A changeover time is required whenever the servers are replaced. In this model, the changeover time of a server is initiated as soon as one server completes the services of the units waiting before him namely at the end of his busy period. It may be noted that the empty state of the system is never reached. A comparison of this model with the model with empty state has been done

Batra, D. P.

English

Defence Science Journal

SINGLE QUEUE ATTENDED BY ALTERNATIVE SERVERS WITH CHANGEOVER TIMES Scientific Analysis Group, New Delhi (Received 19 December 1974; revised 10 December 1975) A single queue is serviced by two servers alternately, in alternate b u g  periods. A changeover time is required whenever the servers are replaced. rn tthie motlel, the ch&ngeover time of a server is initiated as soon as one server completes the services of the units waiting before him namely at the end of his busy period. It may be noted that the empty state of the system is never reached. A comparison of this model With the model with empty state has been done. P Many queuing models involving changeover times and the interaction of queues have been studied in the literature bv Gaverl, Takacsa. Skinners. Leibowitz%nd Tanners. Recentlv Scott6 has considered a queuing model serviced by alternating servers working in shifrs. An advantage in studying such a system is that if redundant modules are provided for service mechanism and if they work alternately, then there is a possibility of providing an efficient repairing or maintenance schedule for the module which are a t  rest without affecting the service provided by the system. Batra and Thirnvengadam' studied the problem of alternating servers with initial changeover times, i.e., when server's switchingover takes place, some initial time is spent in setting up the service mechanism and the server switchover takes place after recurrence of empty state of the system. In  this paper, we have assumed that the changeover time of the server is initiated as soon as the server comdetes ihe service of the units that are wai tk .~  before him. Further. a t  the end of e " changeover time if no u i t  is waitingfor service for a server, then the other changeover time is' again initiated for the other server. It may be noted here that the system will have no idle state. The study of this model helps us to compare the relative performance of the operational characteristics with the model in which the e 6 t y  state of the system is introduced?. - S T A T E M E N T  O F  THE P R O B L E M  Customers arrive in the system according to a Poisson process with intensity h and join in a single counter queue, serviced by two servers namely server 1 and server 2. They work alternately with independent service rates, hence the service times of the units are governed by two probability distributions. Let S; (8) (i=l, 2) be the probability denflity of service time distribution of units when serviced by the server i (i=l, 2) res- pectively. Let rli (8) d19 be the conditional probability that the service completion of a unit takes place bet- ween (8, efde) given tliat it has not been completed upto ,time by the server i. Let 8(+) and S', (+) be the probability densities of changeover time distributions from server 2 to 1 and from server 1 to 2 respectively. Also we define ,,hi(+) d+ (i=1,2) be the conditional probability of changeover time completion between (b and (bf d+ given that i t  has not completed upto time + for the density function S';(#)(i=l, 2). Let us introduce the following definitions of the variables. M(t )  = The number of customers waiting or being serviced at time t. I N(t) = The number of customers already serviced upto time t. ~ ~ ( i  = 1, 2) = The elapsed service time of a customer under service a t  time 6me t with the server i. Tti(i = 1, 2) = The elapsed changeover time at time t when the changeover takes place from service i. Using the method of supplementary variabless-we define the following probabilities, . q; (m, YZ, 8, t) = P, [H( t )  = m, N(t) = n, 9 < TE < 9 + do] (i = 1,2) , The above probabilities are mutually'mclusive and exhaustive and provide a complete markovian characteristics of the system, We relate the state of the system a t  time t + A t  to the state a t  time t ,  t h m  i the differential difference equations connecting these probabilities and governing tbe system are given by for m, 2 0 (5 +-$-+[A + h ( + ) ] } ~ a ( ~ ~ ~ ~ # ~ t )  = A Q I ( * , ~ ~ I , ~ )  farm 2 0  I The above eq&ations are to be solved subject to the following boundary conditions CQ n CQ * ql ( m , i ~ ,  0, t )  =Jql (m + 1, ,a - 1,0, t )  7% (B)  a8 + Ja n, a t )  A (4) Y . (5) 0 0 It may be noted that the range of values for the variable n  in the above probabilities depends upon the initial conditions. Thus, for the initial condition Q1 (m, n, #, o) = Sno a(+) -. The values of n  for the equations (17 to (4) are n  2 0, n  2 1, n  2 0 and n  >, 1 resptively. . T BOLUTIOT dB' THE PROBLEM We define the generating functions as under : 00 PI (0,x, g, t )  = C X r n f 1 *  . (09 9, $1 = C m = 1  m = l  n = O  - \ I BATRA :Single Queue Attended by ~ ~ 6 r n a h ' @ e r v e ~ ,  With the help of these generating functions, after multiplying by the appropriate powers of y and x and summing over, we obtain the following equations : ($ +-3 + [ -A.(;-x) - . -I- 7, ($1 j} BI (8, 9 ,  tr = 0 (9) i f' i , {$ + 5 + [ ~ ( 1 - X J  + 7, (0)  1) p2 (6, X,  y, t )  = Q (10)  a (5 + 3 < [ A  ( 1 5 )  + h (4) I} 4 (4, x, Y, t )  = 0 (11) I ( % + $  +[h(l-%)+$,($)])a,cl,X,Y,.t) = 0 - (12) We define the Laplace transform of a real valued function f ( t )  by - . .  A r) f (.) = f e - 8 t f ( t ) r  - where Re ( s )  2 0 0 " I Applying the Laplace transform with the initial colndaion 7 Qz b, % $9 0 )  = am1 8% 8 ($1 (13) - - to the equati~s'(9f to  (12) spdbkegating the resulting differential equations, we get - .  .. .  - - 4 -[A (1-)+81$ -j$a ($1 @ ' . G$ @, -x, y, s) =-%f2 (e;--x, &-q"e . - . . e -0 (17) Now, we evaluate the values of the functions Fl (0, X, y; a ) ,  FB ( 0 , q  y, a), GI (0,  x, y, s )  ssd Ga (0 ,  x, y, .) aafollows. . - From the definition of generating fundions, the boundary conditions (7) and (8) after the application of Laplace tralisform becoxe - Similarly we can easily get ac $ 5  - E; (0, x,  $7, S )  = y zr. jPlu m+l (6. y. 8 )  (6)  + z x m  JG, (4, y, 8 )  +l (4) a+ (201 0 m=l 0 m=l "0 - The (20) on using (18) after a little simplification becomes .-. m ac $7 J F ~  (6, x ,  I ,  8 )  ( e )  + SG ($1, xy y, 8 )  +I (<Y -3 (0, x, y, 8 )  ($1) PI (0, 2, y, 8 )  = - x 0 0 Sub8tituting (14) and (16) in (21) one can easily obtain _ - Fl (0,  X ,  Y ,  8 )  = [X 4- G, (0, x, y, s ) ]  31' [A  ( 1  - 5) + 81 - P2 (0, X ,  y, sj . (22) 1 - 2. s ; [ X ( l - x )  + s ]  x where pl(s) and &(s)(i = 1,2)  are the Laplace transform of the probability density functions fj'i(#) and &(e) (i = 1,2)  respectively. : Similarly we have With the help of boundary conditions (8) and equations (15)  and (19) where ( a )  and SI ( s )  are defined earlier. Now it  may be noted that denominators of expressions (22) asd (23) has each one root x inside the circle ]a1 = 1. Let US suppose thatfi(y, s) (i=l, 2) be the root of the equation in x lies inside / X I =  1 Noting that 07 (0, x, y, s )  and Gp (0,5,  y, S )  are the functions-that are independent of x and pl(Oy x, y, s) and &(O, x, y, s) are analytic functions of x inside Ixl=l the numerator gf (23) and (24) should also vanish a t  f l (y ,  s)  and f2(y, s) respectively. Therefore, we have G (0, x ,  y, 8 )  = fi (YY 8)  pi [ A  (1-fl (Y,  8 ) )  -4- s ]  [ A  (1-f2 (y ,  8 ) )  + s ]  l - ~ l ~ A { l - f ~  (Y ,  s) 1 + 81' ;F', [ A  (1-f, (y, s ) )  + s ]  and Defining the generating function of the queue length probabilities as - + [ga (0,  x, y, S )  ??a { A ( 1 4 )  + s }  - GI (0 ,  y, 41 [I-& { A ( 1 - 4  4- 811 Y - A (I--) + s .i- - [1  - { A (1-33) d- s } ]  - The Laplace transform of the generating function of queue length distribution irrespective of the number of services completed is given by - ,-' ; (z, [{l+a(~.dy 1 7 ~ ) } 8 <  { ~ ( 1 ~ )  + S  ) - G ( o , x , ~ ,  S,I [ 1 - 8 ; { ~ ( l - - a )  +SIJ I ,I ( 1 - 4  + 8 + [l- $ S; ( A (1 -4  + s )  The steady state distribution of the queue length distribution can be obtainea by taking the limit s-+O - sx(x,l ,s)  = x ( ~ $ 1 )  -. where Therefore, the mean number of units in the system is given by where ,i = Tx , ( x )  de 0 lea 0 pi i= j5 & ($1 0 ( - > t p.jh 5 I' x2 &+( (x) as (i.= 1 ,  2) 0 - The syatem of equatiow (1)to (6) can be solved for any other given initial conditions. . 5 D1SCUS;SION O F  R E S U L T S  a In the earlier paper6 it was assumed. that when first server is replaced by second server a changeover , time is required and the empty stateof the system also exists. The expression for the probab3ity that no unit is present in the system, i.e., the proportion of time the system is idle is given by I 1 - t -Anl '  + 1 +Aaf]-l l - A y 1  A - (34) i The exprassion for the irobabilityfor the proportion of time the sygtem switchover takes place w&hOut servicing a, unit is given by - QO ca p = limit , "'0 [ S s g 1  (4, x, 1,s)  h (4) d+ + 1 8 G2 (4, li 8) (I2 (4) 8 3 0  0 0 1 After substituting the values and simplifying 4 rll' \ . i -> p = [ l - - h q l  + (36) - rlz) " and 8: (A)  = > Assumhg &' ( A )  = ~ + r l l )  A + 72' i.e., the changeover time distribution is exponential . 72' ?- 1 - A %  + 1 - X v z  (37) C Substituting I X qit= pi) (4 = 1, 2) expressions (34) and (37) become respectively 1 + pfl I (38) 1 - Pl (39) In order to study the behaviour of P and H a programme in Portran language was written and executed on IBM 1620 computer. The parameters Xi, pi and pi' (i=l, 2 )  take values from 0.1 to 0.9 with -rate of inorement as 0.1. 166 - - -  , dams : 8inile ~ h n e  ~ttended by Alternative betvote I - -  * C O N C L U ~ I O N  ?he expression for P which represents system chngeover time without servicing a unit dppmds upon the form of changeover time distribution whereas the expreasions for H which reprewnts idle time spent by tlie system without servicing a unit is independent of the form of changeover time distribution. After going through detailed numerical results obtained through computer the following conclusion is made. I n  the- region under consideration, i.e., hi, pi gnd p', (i=l, 2) all take values from 0.1 to 0.9 it has been found that P > H in greater part of this regidn. Therefore, the 'model' in which empty state af the system exists is more efficient than the model considered in this paper. This trend can be easily seen from the table 1/ P-repr~enta the proportion of time the system switchover takes plwe m o u t  @doing s unit. H-reprment~ the proportion of time the ayetern remains idle. 167, '= 

Single Queue Attended by Alternative Servers with Changeover Times

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Single Queue Attended by Alternative Servers with Changeover Times

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Defence Science Journal