Transient Solution of an M/M/c Queue with ...

Information and Management Sciences Volume 12, Number 3, pp.15-27, 2001

Transient Solution of an M/M/c Queue with Heterogeneous Servers and Balking B. Krishna Kumar Anna University India

D. Arivudainambi Anna University India

Abstract This paper analyses a multi-channel Markovian queueing system with heterogeneous servers and balking behaviour. For the system initiated with a random number of customers, the transient solution for the system size distribution is obtained explicitly in a direct way and the steady state probabilities are deduced. Some special cases are also discussed for the case of single server and two server models.

Keywords: System Size, Almost Lower Triangle, Balking, Heterogeneous Servers, Steady State Probabilities.

1. Introduction Queues have been under intensive investigation for several years in a variety of elds. Much of the huge literature on queues is con ned to results describing steady state solutions only. But in many potential applications of queueing theory, the practitioner needs to know how the system will operate up to some instant t. Many systems begin operations and are stopped at some speci ed time t. Businesses or service operations such as rental agencies or physician's oces, which open and close, never operate under steady state conditions. Furthermore, if the system is empty initially, the fraction of time the server is busy and the initial rate of output etc., will be below the steady state values so that the use of steady state results to obtain these measures is not appropriate. Thus, the investigation of the transient behaviours of queueing processes is also important, not only from the view point of the theory but also from the view point of applications. Most of the multiserver queueing model problems tackled in the literature assume the servers to be identical. However, this situation is not very realistic and can prevail only when the service process is highly mechanically controlled. In the case of human servers, Received September 2000; Revised and Accepted January 2001.

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Information and Management Sciences, Vol. 12, No. 3, September, 2001

we cannot expect them to work at the same rate. We face situation of this kind in our everyday life, for example, at checkout counters in grocery stores and departmental stores, in banks, and many others. Morse (1958) seems to be the rst to introduce the concept of heterogeneity in service and obtained the steady state results for the following two cases (i) no queue is allowed before the service facility and (ii) an in nite queue is allowed before the service facility. It is to be pointed out that the explicit expressions for the state probabilities in the second case are not given. Further, Saaty (1960) has discussed Morse's problem and obtained the explicit expressions for the steady state probabilities and the mean number in the system. Sharma and Dass (1988) have also analysed the busy period distributions for M/M/2/N queueing system with heterogeneous servers and obtained the probability density function of the busy period and its mean and variance. Recently, Krishna Kumar et al. (1993) have demonstrated how the transient solution for the state probabilities in a single server Poisson queue with balking can be obtained in a simple and direct way. Exploiting this novel technique, we obtain the transient solution for the probabilities in a multiserver queueing system with `c'heterogeneous servers and balking. An arriving customer may not join the queue, if there are any customer in the system, i.e., the customer may balk. Haight (1957) studied this problem for a single server queue in equilibrium with Poisson input and exponential holding times, for various balking distributions. Singh (1980) has considered two heterogeneous server Markovian queues with balking and compared its eciency with the corresponding homogeneous system. Artalejo and Lopez-Herrero (2000) have analysed an M/G/1 retrial queue with balking and obtained the limiting distribution of the number of customers in the system. Other related work on balking are found in Gaver (1959), Ancker and Gaforian (1963a, b), Hadidi (1974), Natvig (1975), Schellhaas (1983), Ikeda and Nishida (1998) and AbouEl-ata and Hariri (1995). Takacs (1960) considered the transient behaviour of a single server system with Poisson input and arbitrary service time distribution. He assumes that if a customer arrives at an instant when the server is busy then he may or may not join the queue. Here we obtain the transient solution of the system size of an M/M/c queueing system with `c' heterogeneous servers and balking. In the next section, the system of dierential equations for the problem are set up. This system is solved in a direct and elegant way.

Transient Solution of an M/M/c Queue with Heterogeneous Servers and Balking

17

Section 3 considers the steady state distribution. Special cases are discussed in section 4.

2. Model Description and Analysis We consider an M/M/c queueing system under the assumption of Poisson arrivals with rate  and negative exponentially distributed service times with dierent service rate i (i = 1; 2; 3; : : : ; c) for each of the `c' servers. Without loss of generality assume that 1 > 2 > 3 >


> c , which also will imply that rst arriving customer in the queue (when it is empty) joins the 1-st server for service and thereafter the second arriving customer goes to the 2-nd server, which it nds free and so on. If an arriving customer nds all servers busy, then he joins the system with probability p(0 < p < 1) and otherwise balks with probability q(q = 1 ; p). Let fX (t); t > 0g be the number of customers present in the system at time t, Pn (t) = P fX (t) = ng be the probability that there are n customers in the system at time t, n = 0; 1; 2; : : : ; and P (s; t) be the corresponding probability generating function. We assume that there is no customer in the system at t = 0. The system is governed by the following set of dierential-dierence equations: dP0 (t) = ;P (t) +  P (t) (2.1)

dt

0

1 1

n nX +1 dPn(t) = ;( + X  ) P ( t ) + P ( t ) + i Pn+1(t); 1  n  c ; 1 (2.2) i n n ; 1 dt i=1 i=1 c c X dPc (t) = ;(p + X  ) P ( t ) + P ( t ) + i Pc+1(t) i c c;1 dt i=1 i=1

(2.3)

c c X dPn (t) = ;(p + X i )Pn (t) + pPn;1 (t) + iPn+1 (t); n  c + 1 dt i=1 i=1

(2.4)

with Pn (0) = on , the Kronecker symbol. The we have the following proposition:

Proposition 1. The time-dependent state probabilities Pn+c(t) are given by Z t Pn+c (t) = n n e;(p+ )(t;u) In((t (;t ;u)u)) Pc (u)du; n = 1; 2; 3; : : : 0

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Proof. If we de ne P (s; t) = qc(t) + with

1 X n=1

Pn+c(t)sn ; P (s; o) = 1

qc (t) = the system of equations (2.1)-(2.4) yields

c X n=0

c X

(

c @P (s; t) = ps ; (p + X i)+ i=1s @t i=1

Pn(t);

i )

(2.5) (2.6)

[P (s; t) ; qc (t)]+(s ; 1)pPc (t):

(2.7)

The solution of this dierential equation is easily obtained as (

P (s; t) = exp ; [(p + +

Z t

0

c X i=1

i ) + (ps + (

c X i=1

i)=s)]t

fp(s ; 1)Pc (u) ; [ps ; (p + (


exp ; [(p + It is well known that if =

c X i=1

c X i=1

c X

i ) + (ps + (

p

i=1

c X i=1

)

i ) + (

c X

i )=s]qc(u)g i=1 )

i )=s)](t ; u) du:

(2.8)

p

i,  = 2 p and = p= , then

1 X

( sn )In (t) expf;(ps + s )tg = n=;1

where In(
) is the modi ed Bessel function. Using this in (2.8) and comparing the coecient of sn on either side, we get, for n = 1; 2; 3; : : : ;

Pn+c(t) = expf;(p + )tgIn (t) n Z t +p expf;(p + )(t ; u)gPc (u)[In;1 ((t ; u)) n;1 ; In ((t ; u)) n ]du Z t

0

; expf;(p + )(t ; u)gqc(u)[pIn;1 ((t ; u)) n;1 0 ;(p + )In ((t ; u)) n + In+1((t ; u)) n+1 ]du

(2.9)

and for n = 0,

qc(t) = expf;(p + )tgI0 (t) Z t +p expf;(p + )(t ; u)gPc (u)[I1 ((t ; u)) ;1 ; I0 ((t ; u))]du

;

Z t

0

0

expf;(p + )(t ; u)gqc (u)[I1 ((t;u)) ; (p + )I0 ((t;u))]du: (2.10)

Transient Solution of an M/M/c Queue with Heterogeneous Servers and Balking

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As P (s; t) does not contain terms with negative powers of s, the right-hand side of (2.9) with n replaced by ;n, must be zero. Thus Z t

expf;(p + )(t ; u)gqc (u)
[pIn+1 ((t ; u)) n;1 ; (p + )In ((t ; u)) n + In;1((t ; u) n+1 ]du = expf;(p + )tg Z t +p expf;(p + )(t ; u)gPc (u)[In+1 ((t ; u)) n;1 ; In ((t ; u) n ]du (2.11) 0

0

where we have used I;k (
) = Ik (
): Usage of (2.11) in (2.9) considerably simpli es the working and results in elegant expression for Pn (t): This yields, for n = 1; 2; 3; : : : ; Z t n Pn+c(t) = n expf;(p + )(t ; u)g In ((t (;t ;u)u)) Pc(u)du: (2.12) 0 This completes the proof of the proposition 1. Now, the probabilities Pn (t), n = 0; 1; 2; : : : ; c; remain to be found. For this we consider the system of (2.1) and (2.2). Equation (2.1) together with (2.2) can be expressed in the form, dP (t) = AP (t) + P (t)e (2.13) c c dt

where with

P (t) = (P0 (t); P1 (t);


; Pc;1 (t))T ; A = (aij )cc 8 > > > > [ + (1 > > > > > > > j > >X
k=1 k > > > >  > > > > > > > > :0

; oj )

j X k=1

k ] ; j = i; i = 0; 1; 2; 3; : : : ; c ; 1: ; j = i + 1; i = 0; 1; 2; 3; : : : ; c ; 2: ; j = i ; 1; i = 1; 2; 3; : : : ; c ; 1:

(2.14)

; otherwise

and ec is a column vector of order c with 1 in the last place and zero in the remaining places. In the sequel, for any function f (t), let f  (z ) denote its Laplace transform. Now, by taking Laplace transform, the solution of (2.13) is obtained as

P  (z ) = (zI ; A);1 f Pc  (z )ec + P (0)g

(2.15)

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with

P (0) = (1; 0; 0; : : : ; 0)T :

(2.16)

Thus, only Pc  (z ) remains to be found. We observe that if

e = (1; 1; 1; : : : ; 1)T ; eT P (z) + Pc  (z ) = qc (z): Taking Laplace transform and solving for qc (z ), we obtain from (2.10) as q zqc (z) = Pc(z ) 12 f[(z + p + ) ; (z + p + )2 ; 2 ] ;  g + 1: Using (2.18) in (2.17) and simplifying, we get 1 ; zeT (zI ; A);1 P (0) ): Pc (z ) = ( p (z+p+ ); (z+p+ )2;2 T ; 1 ]+ z e (zI ; A) ec z + p ; [ 2

(2.17)

(2.18)

(2.19)

In (2.15) and (2.19), (zI ; A);1 has to be found. For smaller order matrices the usual procedure can be employed. For higher order matrices, we can follow the procedure given by Raju and Bhat (1982), to get the elements of the matrix (zI ; A);1 . To this end, we let (zI ; A);1 = (aij (z ))cc : We note that (zI ; A) is almost lower triangular. Following Raju and Bhat, we obtain, for i = 0; 1; 2; : : : ; c ; 1 8 1 uc;j +1(z )ui;0 (z ) ; ui;j +1(z )uc;0 (z ) ; j = 0; 1; 2; : : : ; c ; 2 > > > > j +1 > uc;0(z) > X > <  k  aij (z ) = > k=1 (2.20) > > > u (z ) ; > > > i;0 j = c ; 1: : uc;0(z) where uij (z ) are recursively given as

uii = 1; ui+1;j =

z++ i+1 X k=1

i X k=1

k

i = 0; 1; 2; : : : ; c ; 1 k

;

i = 0; 1; 2; : : : ; c ; 2

Transient Solution of an M/M/c Queue with Heterogeneous Servers and Balking

ui+1;i;j =

and

(z +  +

i X k=1

k )ui;i;j ; ui;1;i;j i+1 X

k k=1 8 cX ;1 > > > [ z +  + k ]uc;1;j ; uc;2j ; > < k =1 uc;j = > cX ;1 > > > k ; :z +  + k=1 ui;j = 0;

;

21

j  i; i = 1; 2; 3; : : : ; c ; 2: j = 0; 1; 2; : : : ; c ; 2 j = c ; 1:

(2.21)

for other i and j:

We have suppressed the argument z to facilitate computation. The advantage in using these relation is that we do not evaluate any determinant. Using these in (2.19), we obtain

pc (z) =

1;z

z + p ;

"

cX ;1 i=0

ai0 (z)

# p (z +p+ ); (z+p+ )2 ;2

2

+ z

cX ;1 j =0

aj;c;1(z )

(2.22)

and for k = 0; 1; 2; : : : ; c ; 1;

Pk  (z) = ak0 (z ) + akc;1 (z )Pc  (z ):

(2.23)

We observe that aij (z ) are all rational algebraic functions in z . The cofactors of the (i; j )th element of (zI ; A) is a polynomial of degree c ; 1 ; ji ; j j. In particular, the cofactors of the diagonal elements are polynomials in z of degree c ; 1 with the leading coecient equal to 1. In fact uc;0 (z ) = 0 is the characteristic equation of A. Since a00 = ; 6= 0, it is also known that the characteristic roots of A are all distinct and negative (Ledermann and Reuter (1954)). Hence the inverse transform aij (t) of aij  (z ) can be obtained by partial fraction decomposition. Let zi , i = 0; 1; 2; : : : ; c ; 1, be the characteristic roots of A. Then, after partial fraction decomposition and simpli cation, (2.22) becomes

Pc (z ) =

N  (z )

p

z + p + + (z + p + )2 ; 2 [1 ; 2

2 (1p; D (z )) ] z + p + + (z + p + )2 ; 2

(2.24)

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Information and Management Sciences, Vol. 12, No. 3, September, 2001

where

N  (z ) = D (z) = with the constants Ni and Di given by

c;1 X

Ni i=0 z ; zi

cX ;1

Di i=0 z ; zi h

Ni = zlim !z (z ; zi ) 1 ; i

and h

Di = zlim !z (z ; zi ) 1 ; i

(2.25)

cX ;1 i=0 cX ;1 j =0

(2.26) i

zai0 (z )

(2.27) i

ajc;1(z) :

(2.28)

Hence (2.24) simpli es into

1 X n X

(;1)m (n + 1)n+1 (2p)n+1 n=0 m=0 ! p 2 ; 2 ]n+1 n [( z + p +

) ; ( z + p +

)  m  (n + 1)n+1 m (D (z )) which on inversion yields,

Pc (z ) = N (z)

(2.29)

!n+1 m n!  ( ; 1) Pc(t) = m (n + 1) 2p n=0 m=0

1 X n X




Z t

0

Z u

N (t ; u)

0

DC (m) (u ; v) expf;(p + )vg In+1v(v) dudv (2.30)

where DC (m) (t) is the m-fold convolution of D(t) with itself with Dc(0) = (t), the Dirac delta function. Also

Pk (t) = ak0(t) +

Z t

0

akc;1 (u)Pc (t ; u)du ; k = 0; 1; 2; : : : ; c ; 1;

where Pc (u) is given by (2.30). Hence we have the following theorem.

Theorem 1. The transient probabilities Pn(t) for the system size are Pn(t) = an0 (t) +

Z t

0

an;c;1(u)Pc (t ; u)du ; n = 0; 1; 2; : : : ; c;1

Z t n Pn+c(t) = n e;( p+ )(1;u) In((t (;t ;u)u)) Pc (u) 0

; n = 1; 2; 3; : : : ;

(2.31)

Transient Solution of an M/M/c Queue with Heterogeneous Servers and Balking

where

23

!n+1 m n!  ( ; 1) P (t) = m (n + 1) 2p n=0 m=0

1 X n X




Z t

0

N (t ; u)

Z u

0

DC (m) (u ; v)e;(p+ )v In+1v(v) dvdu:

In the next section, the steady state results are derived. Special cases are considered in section 4.

3. Steady State Probabilities Let Pn be the steady-state probability de ned by Pn = limt!1 Pn (t), n = 0; 1; 2; : : :. Then we have the following theorem.

Theorem 2. If p < , then the steady-state probabilities Pn are given by c Y

m=n+1

(1 + 2 +


+ m )

Pc ; c;n Pn+c = p

Pc ; n = 1; 2; 3; : : : ; Pk =

!n

where (

c Y

(1 + 2 +


k ) );1

c;1 k=j +1 X

n = 0; 1; 2; 3; : : : ; c;1

Pc = ; p + j =0

c;j

Proof. It is well known that for steady state p < 1

and

lim P (t) = Pk = zlim zP t!1 k !0 k

From (2.15), we have

P



 (z ): 

= zlim zP  (z) = Pc zlim (zI ; A);1 ec : !0 !0

Also from (2.18), we have

zqc (z) = 1 + Pc (z ) 1



2 z + p + ;

q

(z + p + )2 ; 2

(3.1) 



;  :

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Information and Management Sciences, Vol. 12, No. 3, September, 2001

Taking limit as z ! 0, we get

eT P + Pc = 1 + pp; Pc:

(3.2)

  ;1  ;1 e ( zI ; A ) Pc = 1 ; pp; + eT zlim : c !0

(3.3)

This together with (3.1) yields

Also

eT



lim (zT z !0

; A);1



ec =

cX ;1

uj;0(0) : j =0 uc;0 (0)

(3.4)

From the recursive relations (2.20), we solve for uj;0 (0) and obtain 8 > j > > > > j > Y > > > (1 + 2 + >


k=1

c

 > > > j ; 1 > > Y > > > (1 + 2 + > : k=1




+ k )


+ k )

;

j = 0; 1; 2; : : : ; c ; 1

;

j = c:

(3.5)

Using (3.4) and (3.5) in (3.3), we get

Pc =

(

c Y

cX ;1 k=j +1(1 + 2 +


k ) );1

; p + j =0 c;j

(3.6)

Similarly, for n = 0; 1; 2; 3;


; c ; 1,

c Y

Pn =

(1 + 2 +


m )

m=n+1

Also from (2.10), for n = 1; 2; 3; : : : ;

c;n (

)

n p Pn+c = Pc :

Pc:

(3.7)

(3.8)

Thus (3.6), (3.7) and (3.8) provide the steady state probabilities for the system size.

Transient Solution of an M/M/c Queue with Heterogeneous Servers and Balking

25

4. Special Cases Case (i) c=1. In this case (2.1) and (2.6) give (z + )P0  (z ) = 1 + 1 P1  (z ) and zq1 (z ) = zP0  (z) + zP1  (z): Using these in (2.18) and simplifying, we obtain 1 ; z+z    P1 (z) = p 2 2 z + p ; (z + p + 1) ; (z + p + 1 ) ;  + zz+1 1 (n + 1) 1 X

=

1 n=0

n+1



n+1 (z + )n+1

p

(z + p + 1 ) ; (z + p + 1 )2 ; 2 (n + 1)n+1

n+1

(4.1)

This, on inversion, yields 1 (n + 1) Z t n+1 un expf;ug X 1 P1 (t) =  expf;(p + 1 )(t ; u)g In+1(t(;(tu;) u)) du n +1 n ! 0 1 n=0 (4.2) The other probabilities are given by Z t

P0 (t) = expf;tg + 1 expf;(t ; u)gP1 (u)du 0 and for n = 1; 2; 3; : : : ; Z t Pn+1(t) = n n expf;(p + 1)(t ; u)g In ((t (;t ;u)u)) P1 (u)du 0

Case(ii): c=2 Using the recursive relations (2.20) or directly, we obtain 2 3  ;1 z +  +   1 1 5 (zI ; A);1 = (z +  + 1 )(z + ) ; 1 4  z+ The characteristic roots are p (2 + 1 )2 ; 42 ; ; (2  +  )  1 z0 ; z1 = 2 so that a00 (z) = (z z;1 +z )(+z ;1z ) ; (z z;0 +z )(+z ;1z ) 1 0 1 1 0 0 z +  z +  1 0 a11 (z) = (z ; z )(z ; z ) ; (z ; z )(z ; z ) 0 1  0 0 1  1  1 1 a01 (z) = (z ;1 z ) (z ; z ) ; (z ; z ) 0 1 0 1 and

(4.3) (4.4) (4.5) (4.6)

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Information and Management Sciences, Vol. 12, No. 3, September, 2001



a10 (z) =



1

1



(4.7) (z0 ; z1 ) (z ; z0 ) ; (z ; z1 ) : Substituting these in (2.22) and simplifying, P2 (t) can be explicitly found. Also, for k = 0; 1 ;

Pk (t) = ak0(t) + (1 + 2) where

and

Z t

0

ak1 (u)P2 (t ; u)du

(4.8)

 + 1 expfz tg ; z1 +  + 1 expfz tg a00 (t) = z0 z+ ; 0 1 z0 ; z1 0 z1 a11 (t) = zz0 ;+ z expfz0 tg ; zz1 ;+ z expfz1 tg 0 1 0 1  1 a01 (t) = z ; z [expfz0 tg ; expfz1 tg] 0

1

a10 (t) = z ; z [expfz0 tg ; expfz1 tg]: 0 1 The rest of the probabilities Pn (t); n = 3; 4; 5 : : : ; can be obtained from (2.12).

Thus we have obtained an explicit expression for the probability function of the system size in terms of the modi ed Bessel functions.

5. Conclusion We have proposed a multiserver Markov queueing system with heterogeneous servers. The balking behaviour is incorporated by assuming that an arriving customer joins the queue with probability p. Explicit expression for the transient probabilities Pn (t) are found in a direct way along with steady state solution. The special cases of a single and two server queues are also discussed. This model extends substantially the earlier works available in the literature.

References [1] Abou-El-ata, M. O. and Hariri, A. M. A., Estimation and con dence intervals of the M/M/2/N queue with balking and heterogeneity, American Journal of Mathematics and Management Science, Vol. 15, pp. 35-55, 1995. [2] Ancker, C. J. Jr and Gafarian, A. V., Some queueing problem with balking and reneging-I, Oper. Res., Vol. 11, pp. 88-100, 1963a. [3] Ancker, C. J. Jr and Gafarian, A. V., Some queueing problem with balking and reneging-II, Oper. Res., Vol. 11, pp. 928-937, 1963b.

Transient Solution of an M/M/c Queue with Heterogeneous Servers and Balking

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[4] Artalejo, J. R. and Lopez-Herrero, M. J., On the single server retrial queue with balking, INFOR, Vol. 38, No. 1, pp. 33-50, 2000. [5] Gaver, D. P., Imbedded Markovian chain analysis of waiting line process in continuous time, Ann. Math. Statist., Vol. 30, pp. 698-720, 1959. [6] Hadidi, N., Busy period of queue with state dependent arrival and service rate, J. Appl. Prob., Vol. 11, pp. 842-848, 1974. [7] Haight, F. A., Queueing with balking, -I., Biometrika, Vol. 44, pp. 360-369, 1957. [8] Ikeda, Z. and Nishida, T., M/G/I queue with balking, Mathematica Japanica, Vol. 33, pp. 707-711, 1998. [9] Krishna Kumar, B., Parthasarathy, P. R. and Sharafali, M., Transient solution of an M/M/1 queue with balking, Queueing System, Vol. 13, pp. 441-447, 1993. [10] Ledermann, W. and Reuter, G. E. H., Spectral theory for the dierential equation of simple birth and death processes, Phil. Trans. R. Soc., London, Vol. 246, pp. 321-369, 1954. [11] Morse P. M., Queues, Inventories and Maintenance, Wiley, New York, 1958. [12] Natvig, B., On the transient state probabilities for a queueing model where potential customers are discouraged by queue length, J. Appl. Prob., Vol. 11, pp. 345-354, 1975. [13] Raju, S. N. and Bhat, U. N., A computationally oriented analysis of the G/M/1 queue, Opsearch, Vol. 19, pp. 67-83, 1982. [14] Satty, T. L., Time-dependent solution of the many server Poisson queue, Oper. Res., Vol. 8, pp. 67-83, 1960. [15] Schellhas, H., Computation of the state probabilities in M/G/1 queue with state dependent input and state dependent service, OR Spectrum, Vol. 5, pp. 223-228, 1983. [16] Sharma, D. P. and Dass, S., Multiserver Markovian queues with nite waiting space, Sankhya, Vol. B-50, pp. 328-331, 1998. [17] Singh, V. P., Two-Server Markovian queues with balking: Heterogeneous vs homogeneous servers, Oper, Res., Vol. 18, pp. 145-159, 1980. [18] Takacs, L., The transient behaviour of a single server queueing process with a Poisson input, In: Proc. 4th Berkeley Symposium on Mathematical Statistics and Probability. Vol. II, pp. 535-567, 1960.