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Analysis of multi-server queues with station and server vacations
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER
European Journal of Operational Research 110 (1998) 392406
Theory and Methodology
Analysis of multi-server queues with station and server vacations Xiuli Chao a,,, Yiqiang Q. Zhao b a Department of Industrial and Manufacturing Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA b Department of Mathematics and Statistics, University of Winnipeg, Winnipeg, Man., Canada R3B 2E9
Received 1 May 1996; accepted 1 May 1997
Abstract In this paper, we consider G I / M / c queues with two classes of vacation mechanisms: Station vacation and server vacation. In the first one, all the servers take vacation simultaneously whenever the system becomes empty, and they also return to the system at the same time, i.e., station vacation is a group vacation for all servers. This phenomenon occurs in practice, for example, when the system consists of a set of machines monitored by a single operator, or the system consists of inseparable interconnected parallel machines. In such situations the whole station has to be treated as a single entity for vacation when the system is utilized for a secondary task. For the second class of vacation mechanisms, each server takes its own vacation whenever it completes a service and finds no customers waiting in the queue, which occurs, for instance in the post office, when each server is a relatively independent working unit, and can itself be used for other purposes. For both models, we derive steady state probabilities that have matrix geometric form, and develop computational algorithms to obtain numerical solutions. We also analyze and make comparisons of these models based on numerical observations. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Queueing; Embedded Markov chains; Server and station vacations; Hessenberg matrix; Matrix geometric
solutions
1. Introduction There is an extensive literature on queues with vacations because o f their wide applications in c o m p u t e r and c o m m u n i c a t i o n systems. The readers are referred to the recent surveys o f Doshi (1986, 1990) and Teghem (1986), the m o n o g r a p h o f T a k a g i (1991), as well as the references therein. In almost all these papers, it is assumed that there is a single server in the system. In this article we consider a queueing system with multiple servers. T w o vacation mechanisms are introduced: Station vacation and server vacation. F o r the first one, all the servers take vacations simultaneously. T h a t is, whenever the system is empty, all the servers leave the system for a vacation, and return to the system simultaneously when the vacation is completed. So station vacation is a g r o u p vacation for all servers. This occurs when a system consists o f several * Corresponding author. 0377-2217/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PHS0377-22 17(97)00253- 1
X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
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interconnected machines that are inseparable, or when all the machines are run by a single operator. In this case, if the system (or the operator who runs the system) is used for a secondary task when it becomes empty (or available), all the servers (the operator) will then be utilized to perform a secondary task. During this period of time the system is unavailable to further arrivals to the system, and this is equivalent to taking a station vacation. The second vacation mechanism, we call it server vacation, is encountered even more often in practice. In this case, each server is an independent working unit, and it can take its own vacation upon completing serving a customer and finding no customers waiting. This phenomenon occurs, for instance, in post offices where, when observing an empty queue, a clerk goes for other types of work (sorting, distributing, etc.). We note that if c = 1, i.e. the single server system, the two vacation mechanisms coincide and this case has been considered by Tian et al. (1989). We first consider the G I / M / c queue with station vacation. Both the service times and the vacation times are assumed to be exponentially distributed. We express the transition matrix for the embedded Markov chain as a block-Jacobi form and give a matrix-geometric solution of the arrival-point steady-state queue size probabilities. We then consider the case of server vacations. With the additional assumption of Poisson arrival processes, we express the transition rate matrix as a block-Hessenberg type of matrix. Matrix-geometric solution is obtained for this model as well. Computational algorithms are developed for evaluating the performances for both models, and we analyze and compare these models based on numerical observations. In the queueing literature, the vacation mechanisms considered in this paper are termed multiple vacations. That is, the server or station upon returning from a vacation leaves immediately for another one if the system is empty at that moment. Other vacation mechanisms reported in the literature include single vacation, general vacations, etc. (see for instance, Doshi, 1986). The analysis of those vacations for the G / M / c queues can be conducted similarly along the lines of this paper. The rest of this paper is organized as follows. In Section 2 we describe the models and their embedded Markov chains (and transition matrices). In Section 3, we study the matrix-geometric solution structure of the steady state probabilities and develop computational algorithms. Examples, numerical analysis and discussions are given in Section 4.
2. The models
2.1. Queues with station vacations We first describe the model with station vacation. Customers arrive at the system according to a renewal process with interarrival time distribution F(t) and mean 1/2. There are c parallel servers, and the service times of the customers S1, $2,.. •, are independent and identically distributed exponential random variables with rate p, i.e., G(t) = e(Sn <~ t) = 1 - e -ut,
n = 1,2,...
Every customer requires to be served by one and only one server, and leaves the system once the service is completed. As soon as the system is empty (all servers become idle), the station takes a vacation. Exhaustive service discipline is considered here, but other disciplines can be studied similarly. That is, upon completing a vacation, the station returns to the system, and starts to serve customers, if any, till the system becomes empty; otherwise, the station takes another vacation. The process continues until the station returns and finds customers waiting. The vacation times Vl, V2,..., are assumed to be independent and identically distributed exponential random variables with rate 0, i.e., V(t) =P(Vn <~t)= 1 - e -Or, n = 1,2,...
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X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
Let N(t) be the number of customers in the system at time t. The embedded points are selected at the prearrival epochs. Let z, be the arrival time of the nth customer, and denote by Xn = N ( z , - ) the number of customers right before the nth arrival. Define (~
if the nth arrival occurs during a vacation period, if the nth arrival occurs during a busy period.
Y" --
The state of the system is given by (An, Y,) with state space
{(0, 0), (1,0), (1, 1), (2, 0), (2, 1),... ,}. Note that (0,1) does not exist since the station is always in vacation when the system is empty. Because both the service times and vacation times are exponentially distributed, the process { (X,, Yn), n 1> 1} is a Markov chain. For convenience, let us define oo
k
f (cw) _,,,__.. bk -- J ~ e a r t t ),
(1)
0
0
0
Note that bk is the probability that the number o f customers served during an interarrival time is k, given that there are at least c + k customers initially in the system, and that v~ is the probability that during an interarrival time the number of customers served is k, given that initially the station is on vacation and there are at least c + k customers upon return from vacation. Define the transition probabilities for the Markov chain (X,, Y,) as P =P{Xn+I = j ,
Yn+l =
vlX.
= i, Yn = u},
We now compute the transition probabilities P(i,,),(/,v). Four cases are considered: (i) u = v = O, i.e., system is on vacation upon two consecutive arrivals; (ii) u = v = 1, system is busy upon two consecutive arrivals; (iii) u = 1 and v = O, system is busy on one arrival and is on vacation upon the next arrival; and (iv) u = 0 and v = 1, system is on vacation on one arrival and is busy on the next one. These four cases are considered separately. Case 1: Note that the system state changes from state (i, O) to (i + 1, O) only when the next arrival comes before the station returns from vacation, hence OK)
P(i,0),(i+l,0) = P ( V > A) = /
e -°t dF(t) = f * (0),
(3)
, ]
0
where V and A are generic vacation and interarfival times, respectively, and f*(-) is the Laplace transform for F(.). If the server returns from vacation and completes the service for i + 1 customers before the next customer arrives, the server shall go on vacation again, and the next arrival shall observe state (0, 0). Hence for i + 1 ~
t
P(i,0),(0,0) =
1 - e -"(t-s) 0
);+'
Oe-Os ds dF(t)
0
-- bi,o,
(4)
X.. Chao, Y.Q. Zhao / European Journal of Operational Research I10 (1998) 392-406
and f o r i + l
395
>c,
P(i,o),(o,o) =
=
:i/
e -
o
o
i
1 (i - c)!
o
-
--
c)!
o
i'i o
l
(cPs)'-c e-u~ -- e-U(t-s) d~ c#Oe-°SdsdF(t)
o
(5)
bi,o.
Case 2: F o r i f> c - l, and i + 1 >i j >/c, the transition from (i + 1, 1) to (j, 1) occurs when i + 1 - j customers depart during an interarrival time. Since the departure rate during this period is c/~, by conditioning on the interarrival time and using (1), one obtains P(i,i),O,i)
=
i
J e#t)i+l (i + 1 - j)! e-CUt dF(t)
(6)
= bi+l-j.
0
F o r j ~< i + 1 ~< c, the transition from (i, 1) to (j, 1) occurs when i + 1 - j customers complete service. All the i + 1 customers are served simultaneously, and each one is served at rate p. Hence if the interarrival time is t, each o f the i customers completes service with probability 1 - e-ut. The probability that i + 1 - j o f the i + 1 customers complete service by the end o f the interarrival time is therefore
P(i,1),(j,1) =
i (i +j l)(1-e-ut)~+l-Je-Jut dF(t) 0
=aid
forO~
(7)
~
W h e n j ~ c ~< i + 1, the transition from (/, 1) to (j, 1) passes through two stages. In the first stage, i + 1 - c customers are served. During this a m o u n t o f time, the departure rate is constantly at c#. F o r the second stage, c - j customers are served, and this stage is similar to the previous situation. The time length of the first stage has an Erlang distribution with parameter (i + 1 - c, c#). Its density function is
(c#t) ~-~ Ue- ut -O
"
By conditioning on the length o f the first stage, one obtains
P<',l',"l'
=
,, - c,, 0
i0) [i
=o
''
0
~
e -Jut
]
(cps)i-C(e-~ - e-ut)C-Jc#ds dF(t)
- aij for j <. c <. i + l.
(8)
Case 3: Consider the transition probability f r o m (i, 1) to (0, 0). This occurs when a customer arrives to find i customers in the system, and all the customers leave the system before the next customer arrives. We
X.. Chao, Y.Q. Zhao I European Journal o f Operational Research 110 (1998) 392-406
396
consider two cases. If i + 1 ~< c, all the i + 1 customers are served simultaneously. If the interarrival time is t, then each customer leaves the system before t with probability 1 - e -ut. Hence, by conditioning on the interarrival time, one obtains oo
P(i,l),(0,0) = S (1 - e-ta) i+1 dF(t) 0
--ai,o
for0~
~
(9)
When i + 1 > c, the transition from (i, 0) to (0, 0) passes through two stages. In stage 1, i + 1 - c customers leave the system and they occur before a new customer arrives, during this amount o f time, the departure rate is c#. In stage 2, c customers depart before the next customer arrives. Hence, by conditioning on the lengths o f the interarrival time and first stage, which has an Erlang distribution with parameter (i + 1 - c, c#), we obtain
p(i,l),(o,o)= i j c#xe-c~s~lx~)~c~i(l. - e-~
0
0
i
(i - c)!
1
0
0
=ai,o
_ e_~)CdsdF(t)
'
fori+l
>c.
(10)
Case 4: The transition from (i, 0) to (j, 1) occurs when the station returns before the next customer arrives. When c ~
t
f
f [c~(t_-_s)] ~+1-j
0
0
p(,,o),u,1) = j j
=
(i+ 1 - j ) !
e_C~,(,_s)Oe_O~ds dF(t)
vi+l-j,
(11)
and for j ~< i + 1 ~< c, one has oo
t
( i +j 1) ( l _ e_V(t_s)) i+l-Je_JV(t_S)Oe_OSds dF ( t ) 0
0
--bid
for0~
~
(12)
Finally, f o r j ~< c and i + 1 t> c, P(i,0),(jA) can be computed by first reducing the number of customers to c, and then reducing to j oo
0
t
0
t-s
0
- bi,/ for O <~j <. c <. i + l. In summary, the transition matrix can be written as
(13)
X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
Poo Plo
Pox Pll
PI2
P20
P21
P22
Pc-2,O Pc-l,O
Pc-2,1
Pc-2,2
...
Pc-l,l
Pc-l,2
.-.
Pc-2,c-2 Pc-l,c-2
Pc-2,c-I Pc-l,c-I
(14)
AO
Pc,o
Pc,1
Pc,2
...
Pc+l,l
Pc+l,2
.--
Pc,c-2 Pc,c- I A1 Ao Pc+l,c-2 Pc+l,c-1 A2 A1 AO
.
.
where, by
.
.
Pi,0 =
PoA =
(bi,o~
.
.
.
.,
"-
Pij= (~ '
Pii+l = (f*o
bi vi
(f*(O),bo,,),
i>>. l
\ ai,o ] '
=
.
(3)-(13):
Po,o = (bo,o),
Ai
""
Pc+l,0 .
397
ai,i+ibi'i+l) ,
'
(15)
bid~
i>~ 1, j<<.c-1
aid ] '
l <~i <<.c - 1 ,
i= 1,2,...,
A0 =
(16) '
(17)
b0 "
(18)
We note this transition matrix is a special case of the lower block-Hessenberg type matrices. Based on this special structure, we develop its solution and computational algorithms in Section 3.
2•2• Queues with server vacations There are many practical problems where servers take individual vacations. This means whenever a server completes service and there are no more customers waiting in the queue, the server takes a vacation. A typical example is encountered in the post office. When a clerk completes services and finds no customers waiting, he or she might go to work on a secondary task, say sorting letters. This is what we refer to as the server vacation model. As one may expect, this situation is more complicated than the previous case. For instance, at any time epoch, there might be any number o f servers between 0 and c on vacation. We therefore strengthen the assumptions such that the arrival process is Poisson with rate 2. The service times and vacation times are still assumed to be exponential, with rates # and 0 respectively. The vacations are defined as follows• Whenever a server completes service to a customer, and finds no customer waiting, it takes a vacation. Upon returning from vacation, if the server finds any customer waiting in the queue, then it immediately starts serving customers till the queue is empty• If, however, upon returning from vacation, the queue is still empty, then the server takes another vacation. The process continues until it finds a customer waiting when returning from vacation• Let X(t) be the number of customers in the system at time t, and Y(t) the number of servers working in the system at time t. Then the state of the system is described by Z(t) = (X(t), Y(t)). This is a Markov process with state space
X. Chao, Y. Q Zhao I European Journal of Operational Research 110 (1998) 392-406
398
{(i,j), i = 0 , 1 , 2 , . . . , and j--- 0, 1,...,min{i,c}}. Note that under no situation the number of servers working in system can exceed the number of customers in system. We divide the state space into blocks with the ith block being {(i, 0), (i, 1 ) , . . . , (i, min{i,c})}. For block i, the number of customers, i, is fixed, and the number of working servers goes from 0 to min{i,c}. For example, the first block is (0,0), the second block (1,0) and (1,1), the third block (2,0), (2,1), and (2,2), and so on. The size of the blocks increases to c + 1, and after that each block has the same number of states, c + 1. There are three possible events that may result in a state change from (i, j)(j' ~< i): An arrival event (with rate 2) that causes the state to go to (i + 1,j); a server returning from vacation (with rate ( c - j)OI[i > j], where I[.] is an indicator function) that causes the state to change to (i,j + 1), provided j < i; and a service completion (with rate Jk0 that causes the state to change to (i - 1, j) if i > j and to (i - 1, j - 1) if i = j > 0. Considering all possible values for i and j one obtains the Q matrix for this model as follows:
A;
A°
A~
Ao "o
".
"
Ac_2
O" Ac-2 Ac-l
O =
Ac_ 1
A-~
(19)
AA-61 A + A-
A° A-
A÷
Ao A+ "o ".. ".o *Oo °..
where
#
(20)
= (2, 0),
A ° = (-2),
A° =
'
- 2 - cO 0
co)
-#-2
'
0
2
'
(21)
rO 2/~ A~-=
,
°°o (i - 2)/z 0
0
...
0
( i - 1)/z i#
i=
1,...,c,
(22)
X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
-2 - cO
399
cO -2-
p - ( c - 1)0
( c - 1)0
A0= (c - i + 1)0 - 2 - ip
-). - (i -- l)# - (c -- i + l)0 for i = 1 7 . . . , c -
A+= 2 2 ". 2
(23)
1, 0 0 •
7
i= 17...7c-1,
(24)
0 :0 #
2#
A- =
/ • ..
(25) 7
(c-
c#
-2 - cO
A° =
I
cO -2-#-(c-1)0
) (c-l)0 ".
". -2
-
(c -
, -
0
(26)
0
- 2 - c#
A + = M,
(27)
where I is an identify matrix of dimension c + 1. We note that when i ~< c, A,.- is an (i - 1) × i matrix, A/° is an i x i matrix, and A + is an i x (i + 1) matrix. While all A-, A ° and A ÷ are (c + 1)-dimensional square matrices. The Q matrix given by (19) is both upper and lower Hessenberg type matrix, and it admits a matrix geometric solution. As we shall see in Section 3, this solution has a very special structure that can be characterized by a generic blocking matrix.
3. Solution and algorithms In this section, we provide algorithms for computing the steady state distribution of the models defined earlier. Notice that the transition matrix or the generator matrix for station vacation or server vacation model is of a special lower block-Hessenberg type. This special matrix determines that its steady state probability vector also has a special structure. By Neuts (1981), if Qt0, hi, re2,...) is the steady state probability vector partitioned accordingly, then it must have matrix-geometric form 7tk=rccR k-c,
k=e,c+l,...,
400
X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
where R, for the station vacation model, is the minimal nonnegative solution of the following matrix equation: oo
R = ERiAi~ i=O
where A;'s are given in (18). For the server vacation model, R is the minimal nonnegative solution of R2A - + RA ° + A + = O,
and A-, A° and A + are given by (25)-(27). Note that this relationship is only for ~rc,no+l,..., and they are given in terms of no, where rCc has to be computed via the boundary conditions, that are given together with /[0~ ~ 1 , • • • ~ 7~c-1.
There is extensive research and discussion on solution structures for Markov chains with special transition matrices as those given above, see for example, Neuts, 1981, 1989. In this paper we shall take a slightly different approach. In the following, we first provide an explicit expression of the steady state probability vectors for a lower block-Hessenberg transition matrix and then develop the computational algorithm based on this structure. For simplicity, we focus on discrete time, the solution and the algorithms for continuous time can be similarly obtained. By lower block-Hessenberg, we mean that the matrix is of the form Pl,0 Pl,l p=
P2,o
PI,2
P2,1 P2,z •
.
•
.
P2,3 "..
.
where all blank entries are zero and all P,.j are submatrices of size n; × nj with n~ < c~ and nj < oo. In the following, we assume that P is a transition matrix of an aperiodic ergodic Markov chain. Let (n0, ~l, ~2,...) be the steady state probability vectors partitioned accordingly. Then we have the following expressions: (28)
k >1 0,
nk+l = nkRk+l,
where Rk+l = Pk,k+l ~
U~ik+l= Pk,k+l(I -- Uk+l)-1,
k/> 0
(29)
i=0
with Uk+l = Pk+l,k+J +
P/k+l,
Rj i=k+2 \ j = k + 2
J
k >i o,
(30)
'
and rr0 is determined by: o~
=
k
oZIIRi
k=O i = 0
k,0,
n0
R, e = I,
(31), (32)
i=0
where R0 = I is the identity matrix and e the column vector with all elements being 1. The above result can be obtained based on the censoring technique (for example, see Grassmann and Heyman, 1990); a modification of the proof was provided in Zhao (1994). The following probabilistic
X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
401
interpretations are important. We note that Rk+l = (rk+l (U, V))u, v is the matrix of the expected number of visits to level k + 1 between two successive visits to level k since rk+l (u, v) is the expected number of visits to state (k + 1, v) between two successive visits to level k, given that the process starts in state (k, u). Since the transition matrix is lower block-Hessenberg or skip free to the right, rk+l (u, v) is also the expected number of visits to state (k + 1, v) before entering the set J ~ k, given that the process starts in state (k, u), where k
ni
= U
i=0 w=l
We call Uk+l = (vk+t (u, v))u,v the taboo probability matrix of visiting level k + 1 with the taboo set J ~
R[.l = Ao * ( I - U[n])-I ; until Rt.-ljll < ,.
IIR[.]-
The convergence of R[,] to R can be justified as follows. Consider the corresponding station vacation model with a finite waiting capacity K < oo. The last row block in the transition matrix is identical to the second from the last. The finite model approaches the infinite model as the waiting capacity increases. Therefore, the matrix o f the expected number of visits to level k + 1 between two successive visits to level k for the finite model converges to R as K ~ oo for any fixed k >i c. Those matrices are computed by Algorithm 1. Algorithm 2: Compute Rk for k = 1 , 2 , . . . , c - 1: For k = c - 1 , c - 2 , . . . , 1: compute Uk according to (30); Rk = Pk-l,k * (I -- Uk)-l; R0 = I; it0 = 1/(~~-i=0 c-i I-Ik=0Rk i + {r--tc-I R "~ g ' ~ R i-(c-1)'~" For k = 1 , 2 , . . . , c - 1: ~,llk=0 k) z...,i=c ), 7~k : lrk_lR k.
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All the nonboundary probability vectors are computed by the matrix geometric solution rck+l = rrkR, 1,c,... Before we can implement the above algorithms, all transition probabilities have to be computed according to the formulas provided earlier. However, this work itself could be numerically challenging as we know from the experience of computing transition matrices for the G I / M / c queue. Fortunately, for some common distributions, say the exponential, the deterministic, the uniform, the Erlang or mixed Erlang distributions, the transition probabilities can be satisfactorily approximated using standard numerical methods for integration. For example, the methods based on Gauss-Legendre integration. A detailed discussion is beyond the scope of our main focuses here, readers may refer to Tijms (1986). After the transition matrix was generated, there are two steps in the implementation corresponding to the two algorithms. Since the block size of all building blocks in a station vacation model is always 2 x 2, while sizes of blocks increase as c increases in a server vacation model, the implementation of the algorithms demand more in the case of server vacation. To illustrate the performance of the algorithms, we provide the number o f iterations required for computing R such that the maximum difference between two corresponding entries of R[n] and R[n_ 1] is smaller than 10-12 for the case of server vacation. These numbers indicate that even for a larger number of servers, the algorithms require a small number of iterations for computing R and the change in this number is not so sensitive to changes in the traffic intensity parameter (Table 1). This number increases abruptly as the vacation rate gets smaller than 0.01. For this case, one can reduce the computational effort significantly by changing the time unit of the vacation such that the vacation rate is not so small. For a detailed discussion of time complexity, memory complexity, and other concerns for this type of algorithms, the readers are referred to Grassmann and Heyman (1993) and Latouche (1993), among others. k = c-
4. Examples and discussions It is easy to implement the two algorithms and to provide some numerical results. We have tested them on various examples. However, since there are numerous reports and discussions on related algorithms, here we are more interested in the analysis and the comparison of the models based on numerical evidence. In this section, we mainly concentrate on this issue. We can provide proofs for some properties suggested by numerical results, but we have not found a proof for some others. Assume that the arrival process is Poisson with rate 2. Our first example is the case where there is only one server. Example 1 (c = 1). In this case, two models coincide since there is only one server in the whole station. The steady state probability vectors can be explicitly expressed.
0
-p),
where p = 2 / p , rq = ;r0Rl and 7rk = 7roRIR *-I for k = 2, 3 , . . . , where:
R, Specifically:
R~---
X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
403
Table 1 The number of iterations required for computing R C
p = 0.01 0 = 0.01 0.1
2
5
10
50
100
15 6
12 6
12 6
14 8
17 9
1
4
5
5
5
5
1(~1000
2
2-3
2-5
2-5
2-5
p=0.1 0 = 0,01 0.1
73 14
48 12 12
1 10-1000 p=0.5 0 = 0.01 0.1
37 14
5
6
6
8
7
2-4
2-4
2-4
2-4
87 28 13 3-6
89 29 29 3 11
136 37 12 2-5
136 38 15 2 7
154 40 15 2--7
132 26
110 26
1
8
10
10
10-1000
2-4
2-4
2-4
42 26
131 32
151 33
1
8
10
11
10-1000
2-5
p = 0.99 0 = 0.01 0.1
36
2-4
92 24
p=0.9 0 = 0.01 0.1
32 15
2-5
2~
52 23
123 31
155 36
156 38
1
9
10
12
14
10-1000
2-4
rrk,0=r~0
2-5
,
2-5
2-6
k=l,2,...,
•
-- 2+0
J
For the above example, it is clear that the stationary distribution always exists regardless o f the value o f 0, provided 2 < # is satisfied. It is also easy to see that rtk,0 converges to 0 as 0 ~ c¢ and there is no limiting probability distribution when 0 ~ 0. The interpretation o f the above results is that as 0 ~ c¢ the server will eventually take no vacation since vacation times converge to zero. Therefore, the model will be the standard M/M/1 queue, 7~k,0 0 and rtk,l = (1 -- p)pk; as 0 ~ 0 the server will never come back, retired. Therefore the system will be unstable no matter what the traffic intensity p is. Numerical results have shown that this property remains true for more general cases. In order to provide some comparison between the two models, we consider the following example. For convenience, we use 00 as the vacation rate for the station vacation model and 0 the vacation rate o f each server for the server vacation model. =
X. Chao, Y. Q Zhao I European Journal of Operational Research 110 (1998) 392-406
404
Example 2 (c = 2). By the analysis in the previous sections, we obtain the following. (i) The station vacation model: All probability vectors can also be expressed explicitly after some work. 00(2# - 2) 7z0 = 2#(00 + 2) + 200' 7~1 =
goRl with
RI
=
+00'
'
and rck+l = ~kR, k = 1 , 2 , . . . , with R=
.
~J (ii) The server vacation model." This model is more complicated than the station vacation model since there are more states in the block. We also explicitly solved this model and found the expressions not structurally interesting except for the diagonal elements of R, which are given by: ,l r0,0 - 2 + 2 0 '
rl'l
=
(2 + 2#) - V/(2 + 0 + 2#
#)2 _
42# '
r2,2 = p . A number of observations have been made based on our numerical examples. First, we find that for fixed values of c, 2 and #, the average numbers of customers in the system for both station vacation model and server vacation model, as functions of 00 and 0 respectively, are decreasing. They converge to the average number of customers in the system of the corresponding M I M I c queue with no vacation. This property can be formally proved by using the sample path comparison technique (Stoyan, 1983; Ross, 1983). Actually, one can show that the numbers of customers in the two models are stochastically decreasing in the vacation parameters 00 and 0 and this is true for any c. To make comparisons between the two models, it might be more interesting to assume Oo = cO, and in this case, the numerical examples show the following phenomena for a variety of system parameters: (i) Consider the probability that there are k customers in the system. This probability for the model with station vacation is larger than that for the model with server vacation for all k ~ k0, where k0 depends on the values of system parameters; or Zj ~zkj(station) >1~j gkj(server) for k ~< k0 and ~ j ~kj(station) ~< )-]j ~kd(server) for k > k0. (ii) Let X be the number of customers in the system in equilibrium. Then the average number of customers in the system in equilibrium for the station vacation model is always smaller than that for the server vacation model; or E(X)(station) ~< E(X)(server). For a general case where c > 2, our numerical computations still support the above claims. We remark that (ii) can be concluded from (i). As a matter of fact, by the cut criterion of Karlin and Novikoff (see Proposition 1.5.1 of S toyan, 1983), (i) implies that the number of customers in the station vacation model is less than or equal to that of the server vacation model in stochastic convex ordering. However, we have not been able to give a theoretical proof for (i) or even (ii). The difficulty is partly because there is no explicit formula for the server vacation model for a general case. Nevertheless, we have an explicit solution for the model with station vacation and explicit expressions of the diagonal elements of the matrix R for the model with server vacation, as discussed in Example 3.
X.. Chao, Y.Q. Zhao I European Journal of Operational Research I10 (1998) 392-406
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Example 3 (c > 2). (i) The station vacation model: All probability vectors can also be expressed explicitly, nk = nk_lRk for k = 1 , 2 , . . . , c - 1 and nk = nk_lR for k = c , c + 1 , . . . with: RI =
,
+ 0o
,
Rk=
,
R=
k=2,3,...,c-1,
. c/~.1
Finally, ( F = ¢, +
1 "1 "1 -1
,
where e is the column vector with one for both entries and
['~+oo ~(~+Oo)]
(1- n ) - I = L O~ O0(ccPt'Z2 Jl (ii) The server vacation model: R is upper triangular and the diagonal elements are given by:
ro,o -- 2 + cO' (,~ --I- 0 + tt) - V/()~ -4- 0 qt_/2) 2 _ 42#
2#
rl'l =
rk,k=
'
2 + (c - 1)0 + kl~ - ~/(2 + (c - k)O + k/t)a-4k),# 2kg ,
k= 1,2,...,c.
Acknowledgements The authors are indebted to an anonymous referee for helpful comments on an earlier version of this paper. X. Chao acknowledges that this research was partially supported by the N S F under D D M 9209526, and Y.Q. Zhao acknowledges that this work was supported by Grant No. 0-40-8185 from the Natural Sciences and Engineering Research Council of Canada (NSERC).
References Doshi, B.T., 1986. Queueing systems with vacations: A survey. Queueing Systems 1, 29-66. Doshi, B.T., 1990. Single server queues with vacations. Stochastic Analysis of Computer and Communication Systems. In: Takagi, H. (Ed.), Elsevier, North-Holland, Amsterdam, pp. 217-265.
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Grassmann, W.K., Heyman, D.P., 1990. Equilibrium distribution of block-structured Markov chains with repeating rows. J. Appl. Prob. 27, 557-576. Grassmann, W.K., Heyman, D.P., 1993. Computation of steady-state probabilities for infinite-state Markov chains with repeating rows. ORSA J. Computing 5, 292-303. Latouche, G., 1993. Algorithms for infinite Markov chains with repeating columns. In: Meyer, C.D., Plemmons, R.J. (Eds.), Linear Algebra, Markov Chains and Queueing Models. Springer, New York, pp. 231-265. Neuts, M.F., 1981. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore. Neuts, M.F., 1989. Structured Stochastic Matrices of MIGI1 Type and Their Applications. Marcel Dekker, New York. Ross, S., 1983. Stochastic Processes. Wiley, New York. Stoyan, D., 1983. Comparison Methods for Queues and other Stochastic Models. Wiley, New York. Takagi, H., 1991. Queueing Analysis - A Foundation of Performance Evaluation, vol. 1. Elsevier, Amsterdam. Teghem Jr., J., 1986. Control of the service process in a queueing system. Eur. J. Oper. Res. 23, 141-158. Tian, N., Zhang, D., Cao, C., 1989.The GIIMI1 queue with exponential vacations. Queueing Systems 5, 331-344. Tijms, H.C., 1986. Stochastic Models: An Algorithmic Approach. Wiley, Chichester. Zhao, Y.Q., 1994. Markov chains of lower Hessenberg-type, Working paper, University of Winnipeg.
European Journal of Operational Research 110 (1998) 392406
Theory and Methodology
Analysis of multi-server queues with station and server vacations Xiuli Chao a,,, Yiqiang Q. Zhao b a Department of Industrial and Manufacturing Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA b Department of Mathematics and Statistics, University of Winnipeg, Winnipeg, Man., Canada R3B 2E9
Received 1 May 1996; accepted 1 May 1997
Abstract In this paper, we consider G I / M / c queues with two classes of vacation mechanisms: Station vacation and server vacation. In the first one, all the servers take vacation simultaneously whenever the system becomes empty, and they also return to the system at the same time, i.e., station vacation is a group vacation for all servers. This phenomenon occurs in practice, for example, when the system consists of a set of machines monitored by a single operator, or the system consists of inseparable interconnected parallel machines. In such situations the whole station has to be treated as a single entity for vacation when the system is utilized for a secondary task. For the second class of vacation mechanisms, each server takes its own vacation whenever it completes a service and finds no customers waiting in the queue, which occurs, for instance in the post office, when each server is a relatively independent working unit, and can itself be used for other purposes. For both models, we derive steady state probabilities that have matrix geometric form, and develop computational algorithms to obtain numerical solutions. We also analyze and make comparisons of these models based on numerical observations. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Queueing; Embedded Markov chains; Server and station vacations; Hessenberg matrix; Matrix geometric
solutions
1. Introduction There is an extensive literature on queues with vacations because o f their wide applications in c o m p u t e r and c o m m u n i c a t i o n systems. The readers are referred to the recent surveys o f Doshi (1986, 1990) and Teghem (1986), the m o n o g r a p h o f T a k a g i (1991), as well as the references therein. In almost all these papers, it is assumed that there is a single server in the system. In this article we consider a queueing system with multiple servers. T w o vacation mechanisms are introduced: Station vacation and server vacation. F o r the first one, all the servers take vacations simultaneously. T h a t is, whenever the system is empty, all the servers leave the system for a vacation, and return to the system simultaneously when the vacation is completed. So station vacation is a g r o u p vacation for all servers. This occurs when a system consists o f several * Corresponding author. 0377-2217/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PHS0377-22 17(97)00253- 1
X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
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interconnected machines that are inseparable, or when all the machines are run by a single operator. In this case, if the system (or the operator who runs the system) is used for a secondary task when it becomes empty (or available), all the servers (the operator) will then be utilized to perform a secondary task. During this period of time the system is unavailable to further arrivals to the system, and this is equivalent to taking a station vacation. The second vacation mechanism, we call it server vacation, is encountered even more often in practice. In this case, each server is an independent working unit, and it can take its own vacation upon completing serving a customer and finding no customers waiting. This phenomenon occurs, for instance, in post offices where, when observing an empty queue, a clerk goes for other types of work (sorting, distributing, etc.). We note that if c = 1, i.e. the single server system, the two vacation mechanisms coincide and this case has been considered by Tian et al. (1989). We first consider the G I / M / c queue with station vacation. Both the service times and the vacation times are assumed to be exponentially distributed. We express the transition matrix for the embedded Markov chain as a block-Jacobi form and give a matrix-geometric solution of the arrival-point steady-state queue size probabilities. We then consider the case of server vacations. With the additional assumption of Poisson arrival processes, we express the transition rate matrix as a block-Hessenberg type of matrix. Matrix-geometric solution is obtained for this model as well. Computational algorithms are developed for evaluating the performances for both models, and we analyze and compare these models based on numerical observations. In the queueing literature, the vacation mechanisms considered in this paper are termed multiple vacations. That is, the server or station upon returning from a vacation leaves immediately for another one if the system is empty at that moment. Other vacation mechanisms reported in the literature include single vacation, general vacations, etc. (see for instance, Doshi, 1986). The analysis of those vacations for the G / M / c queues can be conducted similarly along the lines of this paper. The rest of this paper is organized as follows. In Section 2 we describe the models and their embedded Markov chains (and transition matrices). In Section 3, we study the matrix-geometric solution structure of the steady state probabilities and develop computational algorithms. Examples, numerical analysis and discussions are given in Section 4.
2. The models
2.1. Queues with station vacations We first describe the model with station vacation. Customers arrive at the system according to a renewal process with interarrival time distribution F(t) and mean 1/2. There are c parallel servers, and the service times of the customers S1, $2,.. •, are independent and identically distributed exponential random variables with rate p, i.e., G(t) = e(Sn <~ t) = 1 - e -ut,
n = 1,2,...
Every customer requires to be served by one and only one server, and leaves the system once the service is completed. As soon as the system is empty (all servers become idle), the station takes a vacation. Exhaustive service discipline is considered here, but other disciplines can be studied similarly. That is, upon completing a vacation, the station returns to the system, and starts to serve customers, if any, till the system becomes empty; otherwise, the station takes another vacation. The process continues until the station returns and finds customers waiting. The vacation times Vl, V2,..., are assumed to be independent and identically distributed exponential random variables with rate 0, i.e., V(t) =P(Vn <~t)= 1 - e -Or, n = 1,2,...
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X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
Let N(t) be the number of customers in the system at time t. The embedded points are selected at the prearrival epochs. Let z, be the arrival time of the nth customer, and denote by Xn = N ( z , - ) the number of customers right before the nth arrival. Define (~
if the nth arrival occurs during a vacation period, if the nth arrival occurs during a busy period.
Y" --
The state of the system is given by (An, Y,) with state space
{(0, 0), (1,0), (1, 1), (2, 0), (2, 1),... ,}. Note that (0,1) does not exist since the station is always in vacation when the system is empty. Because both the service times and vacation times are exponentially distributed, the process { (X,, Yn), n 1> 1} is a Markov chain. For convenience, let us define oo
k
f (cw) _,,,__.. bk -- J ~ e a r t t ),
(1)
0
0
0
Note that bk is the probability that the number o f customers served during an interarrival time is k, given that there are at least c + k customers initially in the system, and that v~ is the probability that during an interarrival time the number of customers served is k, given that initially the station is on vacation and there are at least c + k customers upon return from vacation. Define the transition probabilities for the Markov chain (X,, Y,) as P
Yn+l =
vlX.
= i, Yn = u},
We now compute the transition probabilities P(i,,),(/,v). Four cases are considered: (i) u = v = O, i.e., system is on vacation upon two consecutive arrivals; (ii) u = v = 1, system is busy upon two consecutive arrivals; (iii) u = 1 and v = O, system is busy on one arrival and is on vacation upon the next arrival; and (iv) u = 0 and v = 1, system is on vacation on one arrival and is busy on the next one. These four cases are considered separately. Case 1: Note that the system state changes from state (i, O) to (i + 1, O) only when the next arrival comes before the station returns from vacation, hence OK)
P(i,0),(i+l,0) = P ( V > A) = /
e -°t dF(t) = f * (0),
(3)
, ]
0
where V and A are generic vacation and interarfival times, respectively, and f*(-) is the Laplace transform for F(.). If the server returns from vacation and completes the service for i + 1 customers before the next customer arrives, the server shall go on vacation again, and the next arrival shall observe state (0, 0). Hence for i + 1 ~
t
P(i,0),(0,0) =
1 - e -"(t-s) 0
);+'
Oe-Os ds dF(t)
0
-- bi,o,
(4)
X.. Chao, Y.Q. Zhao / European Journal of Operational Research I10 (1998) 392-406
and f o r i + l
395
>c,
P(i,o),(o,o) =
=
:i/
e -
o
o
i
1 (i - c)!
o
-
--
c)!
o
i'i o
l
(cPs)'-c e-u~ -- e-U(t-s) d~ c#Oe-°SdsdF(t)
o
(5)
bi,o.
Case 2: F o r i f> c - l, and i + 1 >i j >/c, the transition from (i + 1, 1) to (j, 1) occurs when i + 1 - j customers depart during an interarrival time. Since the departure rate during this period is c/~, by conditioning on the interarrival time and using (1), one obtains P(i,i),O,i)
=
i
J e#t)i+l (i + 1 - j)! e-CUt dF(t)
(6)
= bi+l-j.
0
F o r j ~< i + 1 ~< c, the transition from (i, 1) to (j, 1) occurs when i + 1 - j customers complete service. All the i + 1 customers are served simultaneously, and each one is served at rate p. Hence if the interarrival time is t, each o f the i customers completes service with probability 1 - e-ut. The probability that i + 1 - j o f the i + 1 customers complete service by the end o f the interarrival time is therefore
P(i,1),(j,1) =
i (i +j l)(1-e-ut)~+l-Je-Jut dF(t) 0
=aid
forO~
(7)
~
W h e n j ~ c ~< i + 1, the transition from (/, 1) to (j, 1) passes through two stages. In the first stage, i + 1 - c customers are served. During this a m o u n t o f time, the departure rate is constantly at c#. F o r the second stage, c - j customers are served, and this stage is similar to the previous situation. The time length of the first stage has an Erlang distribution with parameter (i + 1 - c, c#). Its density function is
(c#t) ~-~ Ue- ut -O
"
By conditioning on the length o f the first stage, one obtains
P<',l',"l'
=
,, - c,, 0
i0) [i
=o
''
0
~
e -Jut
]
(cps)i-C(e-~ - e-ut)C-Jc#ds dF(t)
- aij for j <. c <. i + l.
(8)
Case 3: Consider the transition probability f r o m (i, 1) to (0, 0). This occurs when a customer arrives to find i customers in the system, and all the customers leave the system before the next customer arrives. We
X.. Chao, Y.Q. Zhao I European Journal o f Operational Research 110 (1998) 392-406
396
consider two cases. If i + 1 ~< c, all the i + 1 customers are served simultaneously. If the interarrival time is t, then each customer leaves the system before t with probability 1 - e -ut. Hence, by conditioning on the interarrival time, one obtains oo
P(i,l),(0,0) = S (1 - e-ta) i+1 dF(t) 0
--ai,o
for0~
~
(9)
When i + 1 > c, the transition from (i, 0) to (0, 0) passes through two stages. In stage 1, i + 1 - c customers leave the system and they occur before a new customer arrives, during this amount o f time, the departure rate is c#. In stage 2, c customers depart before the next customer arrives. Hence, by conditioning on the lengths o f the interarrival time and first stage, which has an Erlang distribution with parameter (i + 1 - c, c#), we obtain
p(i,l),(o,o)= i j c#xe-c~s~lx~)~c~i(l. - e-~
0
0
i
(i - c)!
1
0
0
=ai,o
_ e_~)CdsdF(t)
'
fori+l
>c.
(10)
Case 4: The transition from (i, 0) to (j, 1) occurs when the station returns before the next customer arrives. When c ~
t
f
f [c~(t_-_s)] ~+1-j
0
0
p(,,o),u,1) = j j
=
(i+ 1 - j ) !
e_C~,(,_s)Oe_O~ds dF(t)
vi+l-j,
(11)
and for j ~< i + 1 ~< c, one has oo
t
( i +j 1) ( l _ e_V(t_s)) i+l-Je_JV(t_S)Oe_OSds dF ( t ) 0
0
--bid
for0~
~
(12)
Finally, f o r j ~< c and i + 1 t> c, P(i,0),(jA) can be computed by first reducing the number of customers to c, and then reducing to j oo
0
t
0
t-s
0
- bi,/ for O <~j <. c <. i + l. In summary, the transition matrix can be written as
(13)
X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
Poo Plo
Pox Pll
PI2
P20
P21
P22
Pc-2,O Pc-l,O
Pc-2,1
Pc-2,2
...
Pc-l,l
Pc-l,2
.-.
Pc-2,c-2 Pc-l,c-2
Pc-2,c-I Pc-l,c-I
(14)
AO
Pc,o
Pc,1
Pc,2
...
Pc+l,l
Pc+l,2
.--
Pc,c-2 Pc,c- I A1 Ao Pc+l,c-2 Pc+l,c-1 A2 A1 AO
.
.
where, by
.
.
Pi,0 =
PoA =
(bi,o~
.
.
.
.,
"-
Pij= (~ '
Pii+l = (f*o
bi vi
(f*(O),bo,,),
i>>. l
\ ai,o ] '
=
.
(3)-(13):
Po,o = (bo,o),
Ai
""
Pc+l,0 .
397
ai,i+ibi'i+l) ,
'
(15)
bid~
i>~ 1, j<<.c-1
aid ] '
l <~i <<.c - 1 ,
i= 1,2,...,
A0 =
(16) '
(17)
b0 "
(18)
We note this transition matrix is a special case of the lower block-Hessenberg type matrices. Based on this special structure, we develop its solution and computational algorithms in Section 3.
2•2• Queues with server vacations There are many practical problems where servers take individual vacations. This means whenever a server completes service and there are no more customers waiting in the queue, the server takes a vacation. A typical example is encountered in the post office. When a clerk completes services and finds no customers waiting, he or she might go to work on a secondary task, say sorting letters. This is what we refer to as the server vacation model. As one may expect, this situation is more complicated than the previous case. For instance, at any time epoch, there might be any number o f servers between 0 and c on vacation. We therefore strengthen the assumptions such that the arrival process is Poisson with rate 2. The service times and vacation times are still assumed to be exponential, with rates # and 0 respectively. The vacations are defined as follows• Whenever a server completes service to a customer, and finds no customer waiting, it takes a vacation. Upon returning from vacation, if the server finds any customer waiting in the queue, then it immediately starts serving customers till the queue is empty• If, however, upon returning from vacation, the queue is still empty, then the server takes another vacation. The process continues until it finds a customer waiting when returning from vacation• Let X(t) be the number of customers in the system at time t, and Y(t) the number of servers working in the system at time t. Then the state of the system is described by Z(t) = (X(t), Y(t)). This is a Markov process with state space
X. Chao, Y. Q Zhao I European Journal of Operational Research 110 (1998) 392-406
398
{(i,j), i = 0 , 1 , 2 , . . . , and j--- 0, 1,...,min{i,c}}. Note that under no situation the number of servers working in system can exceed the number of customers in system. We divide the state space into blocks with the ith block being {(i, 0), (i, 1 ) , . . . , (i, min{i,c})}. For block i, the number of customers, i, is fixed, and the number of working servers goes from 0 to min{i,c}. For example, the first block is (0,0), the second block (1,0) and (1,1), the third block (2,0), (2,1), and (2,2), and so on. The size of the blocks increases to c + 1, and after that each block has the same number of states, c + 1. There are three possible events that may result in a state change from (i, j)(j' ~< i): An arrival event (with rate 2) that causes the state to go to (i + 1,j); a server returning from vacation (with rate ( c - j)OI[i > j], where I[.] is an indicator function) that causes the state to change to (i,j + 1), provided j < i; and a service completion (with rate Jk0 that causes the state to change to (i - 1, j) if i > j and to (i - 1, j - 1) if i = j > 0. Considering all possible values for i and j one obtains the Q matrix for this model as follows:
A;
A°
A~
Ao "o
".
"
Ac_2
O" Ac-2 Ac-l
O =
Ac_ 1
A-~
(19)
AA-61 A + A-
A° A-
A÷
Ao A+ "o ".. ".o *Oo °..
where
#
(20)
= (2, 0),
A ° = (-2),
A° =
'
- 2 - cO 0
co)
-#-2
'
0
2
'
(21)
rO 2/~ A~-=
,
°°o (i - 2)/z 0
0
...
0
( i - 1)/z i#
i=
1,...,c,
(22)
X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
-2 - cO
399
cO -2-
p - ( c - 1)0
( c - 1)0
A0= (c - i + 1)0 - 2 - ip
-). - (i -- l)# - (c -- i + l)0 for i = 1 7 . . . , c -
A+= 2 2 ". 2
(23)
1, 0 0 •
7
i= 17...7c-1,
(24)
0 :0 #
2#
A- =
/ • ..
(25) 7
(c-
c#
-2 - cO
A° =
I
cO -2-#-(c-1)0
) (c-l)0 ".
". -2
-
(c -
, -
0
(26)
0
- 2 - c#
A + = M,
(27)
where I is an identify matrix of dimension c + 1. We note that when i ~< c, A,.- is an (i - 1) × i matrix, A/° is an i x i matrix, and A + is an i x (i + 1) matrix. While all A-, A ° and A ÷ are (c + 1)-dimensional square matrices. The Q matrix given by (19) is both upper and lower Hessenberg type matrix, and it admits a matrix geometric solution. As we shall see in Section 3, this solution has a very special structure that can be characterized by a generic blocking matrix.
3. Solution and algorithms In this section, we provide algorithms for computing the steady state distribution of the models defined earlier. Notice that the transition matrix or the generator matrix for station vacation or server vacation model is of a special lower block-Hessenberg type. This special matrix determines that its steady state probability vector also has a special structure. By Neuts (1981), if Qt0, hi, re2,...) is the steady state probability vector partitioned accordingly, then it must have matrix-geometric form 7tk=rccR k-c,
k=e,c+l,...,
400
X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
where R, for the station vacation model, is the minimal nonnegative solution of the following matrix equation: oo
R = ERiAi~ i=O
where A;'s are given in (18). For the server vacation model, R is the minimal nonnegative solution of R2A - + RA ° + A + = O,
and A-, A° and A + are given by (25)-(27). Note that this relationship is only for ~rc,no+l,..., and they are given in terms of no, where rCc has to be computed via the boundary conditions, that are given together with /[0~ ~ 1 , • • • ~ 7~c-1.
There is extensive research and discussion on solution structures for Markov chains with special transition matrices as those given above, see for example, Neuts, 1981, 1989. In this paper we shall take a slightly different approach. In the following, we first provide an explicit expression of the steady state probability vectors for a lower block-Hessenberg transition matrix and then develop the computational algorithm based on this structure. For simplicity, we focus on discrete time, the solution and the algorithms for continuous time can be similarly obtained. By lower block-Hessenberg, we mean that the matrix is of the form Pl,0 Pl,l p=
P2,o
PI,2
P2,1 P2,z •
.
•
.
P2,3 "..
.
where all blank entries are zero and all P,.j are submatrices of size n; × nj with n~ < c~ and nj < oo. In the following, we assume that P is a transition matrix of an aperiodic ergodic Markov chain. Let (n0, ~l, ~2,...) be the steady state probability vectors partitioned accordingly. Then we have the following expressions: (28)
k >1 0,
nk+l = nkRk+l,
where Rk+l = Pk,k+l ~
U~ik+l= Pk,k+l(I -- Uk+l)-1,
k/> 0
(29)
i=0
with Uk+l = Pk+l,k+J +
P/k+l,
Rj i=k+2 \ j = k + 2
J
k >i o,
(30)
'
and rr0 is determined by: o~
=
k
oZIIRi
k=O i = 0
k,0,
n0
R, e = I,
(31), (32)
i=0
where R0 = I is the identity matrix and e the column vector with all elements being 1. The above result can be obtained based on the censoring technique (for example, see Grassmann and Heyman, 1990); a modification of the proof was provided in Zhao (1994). The following probabilistic
X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
401
interpretations are important. We note that Rk+l = (rk+l (U, V))u, v is the matrix of the expected number of visits to level k + 1 between two successive visits to level k since rk+l (u, v) is the expected number of visits to state (k + 1, v) between two successive visits to level k, given that the process starts in state (k, u). Since the transition matrix is lower block-Hessenberg or skip free to the right, rk+l (u, v) is also the expected number of visits to state (k + 1, v) before entering the set J ~ k, given that the process starts in state (k, u), where k
ni
= U
i=0 w=l
We call Uk+l = (vk+t (u, v))u,v the taboo probability matrix of visiting level k + 1 with the taboo set J ~
R[.l = Ao * ( I - U[n])-I ; until Rt.-ljll < ,.
IIR[.]-
The convergence of R[,] to R can be justified as follows. Consider the corresponding station vacation model with a finite waiting capacity K < oo. The last row block in the transition matrix is identical to the second from the last. The finite model approaches the infinite model as the waiting capacity increases. Therefore, the matrix o f the expected number of visits to level k + 1 between two successive visits to level k for the finite model converges to R as K ~ oo for any fixed k >i c. Those matrices are computed by Algorithm 1. Algorithm 2: Compute Rk for k = 1 , 2 , . . . , c - 1: For k = c - 1 , c - 2 , . . . , 1: compute Uk according to (30); Rk = Pk-l,k * (I -- Uk)-l; R0 = I; it0 = 1/(~~-i=0 c-i I-Ik=0Rk i + {r--tc-I R "~ g ' ~ R i-(c-1)'~" For k = 1 , 2 , . . . , c - 1: ~,llk=0 k) z...,i=c ), 7~k : lrk_lR k.
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All the nonboundary probability vectors are computed by the matrix geometric solution rck+l = rrkR, 1,c,... Before we can implement the above algorithms, all transition probabilities have to be computed according to the formulas provided earlier. However, this work itself could be numerically challenging as we know from the experience of computing transition matrices for the G I / M / c queue. Fortunately, for some common distributions, say the exponential, the deterministic, the uniform, the Erlang or mixed Erlang distributions, the transition probabilities can be satisfactorily approximated using standard numerical methods for integration. For example, the methods based on Gauss-Legendre integration. A detailed discussion is beyond the scope of our main focuses here, readers may refer to Tijms (1986). After the transition matrix was generated, there are two steps in the implementation corresponding to the two algorithms. Since the block size of all building blocks in a station vacation model is always 2 x 2, while sizes of blocks increase as c increases in a server vacation model, the implementation of the algorithms demand more in the case of server vacation. To illustrate the performance of the algorithms, we provide the number o f iterations required for computing R such that the maximum difference between two corresponding entries of R[n] and R[n_ 1] is smaller than 10-12 for the case of server vacation. These numbers indicate that even for a larger number of servers, the algorithms require a small number of iterations for computing R and the change in this number is not so sensitive to changes in the traffic intensity parameter (Table 1). This number increases abruptly as the vacation rate gets smaller than 0.01. For this case, one can reduce the computational effort significantly by changing the time unit of the vacation such that the vacation rate is not so small. For a detailed discussion of time complexity, memory complexity, and other concerns for this type of algorithms, the readers are referred to Grassmann and Heyman (1993) and Latouche (1993), among others. k = c-
4. Examples and discussions It is easy to implement the two algorithms and to provide some numerical results. We have tested them on various examples. However, since there are numerous reports and discussions on related algorithms, here we are more interested in the analysis and the comparison of the models based on numerical evidence. In this section, we mainly concentrate on this issue. We can provide proofs for some properties suggested by numerical results, but we have not found a proof for some others. Assume that the arrival process is Poisson with rate 2. Our first example is the case where there is only one server. Example 1 (c = 1). In this case, two models coincide since there is only one server in the whole station. The steady state probability vectors can be explicitly expressed.
0
-p),
where p = 2 / p , rq = ;r0Rl and 7rk = 7roRIR *-I for k = 2, 3 , . . . , where:
R, Specifically:
R~---
X. Chao, Y.Q. Zhao I European Journal of Operational Research 110 (1998) 392-406
403
Table 1 The number of iterations required for computing R C
p = 0.01 0 = 0.01 0.1
2
5
10
50
100
15 6
12 6
12 6
14 8
17 9
1
4
5
5
5
5
1(~1000
2
2-3
2-5
2-5
2-5
p=0.1 0 = 0,01 0.1
73 14
48 12 12
1 10-1000 p=0.5 0 = 0.01 0.1
37 14
5
6
6
8
7
2-4
2-4
2-4
2-4
87 28 13 3-6
89 29 29 3 11
136 37 12 2-5
136 38 15 2 7
154 40 15 2--7
132 26
110 26
1
8
10
10
10-1000
2-4
2-4
2-4
42 26
131 32
151 33
1
8
10
11
10-1000
2-5
p = 0.99 0 = 0.01 0.1
36
2-4
92 24
p=0.9 0 = 0.01 0.1
32 15
2-5
2~
52 23
123 31
155 36
156 38
1
9
10
12
14
10-1000
2-4
rrk,0=r~0
2-5
,
2-5
2-6
k=l,2,...,
•
-- 2+0
J
For the above example, it is clear that the stationary distribution always exists regardless o f the value o f 0, provided 2 < # is satisfied. It is also easy to see that rtk,0 converges to 0 as 0 ~ c¢ and there is no limiting probability distribution when 0 ~ 0. The interpretation o f the above results is that as 0 ~ c¢ the server will eventually take no vacation since vacation times converge to zero. Therefore, the model will be the standard M/M/1 queue, 7~k,0 0 and rtk,l = (1 -- p)pk; as 0 ~ 0 the server will never come back, retired. Therefore the system will be unstable no matter what the traffic intensity p is. Numerical results have shown that this property remains true for more general cases. In order to provide some comparison between the two models, we consider the following example. For convenience, we use 00 as the vacation rate for the station vacation model and 0 the vacation rate o f each server for the server vacation model. =
X. Chao, Y. Q Zhao I European Journal of Operational Research 110 (1998) 392-406
404
Example 2 (c = 2). By the analysis in the previous sections, we obtain the following. (i) The station vacation model: All probability vectors can also be expressed explicitly after some work. 00(2# - 2) 7z0 = 2#(00 + 2) + 200' 7~1 =
goRl with
RI
=
+00'
'
and rck+l = ~kR, k = 1 , 2 , . . . , with R=
.
~J (ii) The server vacation model." This model is more complicated than the station vacation model since there are more states in the block. We also explicitly solved this model and found the expressions not structurally interesting except for the diagonal elements of R, which are given by: ,l r0,0 - 2 + 2 0 '
rl'l
=
(2 + 2#) - V/(2 + 0 + 2#
#)2 _
42# '
r2,2 = p . A number of observations have been made based on our numerical examples. First, we find that for fixed values of c, 2 and #, the average numbers of customers in the system for both station vacation model and server vacation model, as functions of 00 and 0 respectively, are decreasing. They converge to the average number of customers in the system of the corresponding M I M I c queue with no vacation. This property can be formally proved by using the sample path comparison technique (Stoyan, 1983; Ross, 1983). Actually, one can show that the numbers of customers in the two models are stochastically decreasing in the vacation parameters 00 and 0 and this is true for any c. To make comparisons between the two models, it might be more interesting to assume Oo = cO, and in this case, the numerical examples show the following phenomena for a variety of system parameters: (i) Consider the probability that there are k customers in the system. This probability for the model with station vacation is larger than that for the model with server vacation for all k ~ k0, where k0 depends on the values of system parameters; or Zj ~zkj(station) >1~j gkj(server) for k ~< k0 and ~ j ~kj(station) ~< )-]j ~kd(server) for k > k0. (ii) Let X be the number of customers in the system in equilibrium. Then the average number of customers in the system in equilibrium for the station vacation model is always smaller than that for the server vacation model; or E(X)(station) ~< E(X)(server). For a general case where c > 2, our numerical computations still support the above claims. We remark that (ii) can be concluded from (i). As a matter of fact, by the cut criterion of Karlin and Novikoff (see Proposition 1.5.1 of S toyan, 1983), (i) implies that the number of customers in the station vacation model is less than or equal to that of the server vacation model in stochastic convex ordering. However, we have not been able to give a theoretical proof for (i) or even (ii). The difficulty is partly because there is no explicit formula for the server vacation model for a general case. Nevertheless, we have an explicit solution for the model with station vacation and explicit expressions of the diagonal elements of the matrix R for the model with server vacation, as discussed in Example 3.
X.. Chao, Y.Q. Zhao I European Journal of Operational Research I10 (1998) 392-406
405
Example 3 (c > 2). (i) The station vacation model: All probability vectors can also be expressed explicitly, nk = nk_lRk for k = 1 , 2 , . . . , c - 1 and nk = nk_lR for k = c , c + 1 , . . . with: RI =
,
+ 0o
,
Rk=
,
R=
k=2,3,...,c-1,
. c/~.1
Finally, ( F = ¢, +
1 "1 "1 -1
,
where e is the column vector with one for both entries and
['~+oo ~(~+Oo)]
(1- n ) - I = L O~ O0(ccPt'Z2 Jl (ii) The server vacation model: R is upper triangular and the diagonal elements are given by:
ro,o -- 2 + cO' (,~ --I- 0 + tt) - V/()~ -4- 0 qt_/2) 2 _ 42#
2#
rl'l =
rk,k=
'
2 + (c - 1)0 + kl~ - ~/(2 + (c - k)O + k/t)a-4k),# 2kg ,
k= 1,2,...,c.
Acknowledgements The authors are indebted to an anonymous referee for helpful comments on an earlier version of this paper. X. Chao acknowledges that this research was partially supported by the N S F under D D M 9209526, and Y.Q. Zhao acknowledges that this work was supported by Grant No. 0-40-8185 from the Natural Sciences and Engineering Research Council of Canada (NSERC).
References Doshi, B.T., 1986. Queueing systems with vacations: A survey. Queueing Systems 1, 29-66. Doshi, B.T., 1990. Single server queues with vacations. Stochastic Analysis of Computer and Communication Systems. In: Takagi, H. (Ed.), Elsevier, North-Holland, Amsterdam, pp. 217-265.
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Grassmann, W.K., Heyman, D.P., 1990. Equilibrium distribution of block-structured Markov chains with repeating rows. J. Appl. Prob. 27, 557-576. Grassmann, W.K., Heyman, D.P., 1993. Computation of steady-state probabilities for infinite-state Markov chains with repeating rows. ORSA J. Computing 5, 292-303. Latouche, G., 1993. Algorithms for infinite Markov chains with repeating columns. In: Meyer, C.D., Plemmons, R.J. (Eds.), Linear Algebra, Markov Chains and Queueing Models. Springer, New York, pp. 231-265. Neuts, M.F., 1981. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore. Neuts, M.F., 1989. Structured Stochastic Matrices of MIGI1 Type and Their Applications. Marcel Dekker, New York. Ross, S., 1983. Stochastic Processes. Wiley, New York. Stoyan, D., 1983. Comparison Methods for Queues and other Stochastic Models. Wiley, New York. Takagi, H., 1991. Queueing Analysis - A Foundation of Performance Evaluation, vol. 1. Elsevier, Amsterdam. Teghem Jr., J., 1986. Control of the service process in a queueing system. Eur. J. Oper. Res. 23, 141-158. Tian, N., Zhang, D., Cao, C., 1989.The GIIMI1 queue with exponential vacations. Queueing Systems 5, 331-344. Tijms, H.C., 1986. Stochastic Models: An Algorithmic Approach. Wiley, Chichester. Zhao, Y.Q., 1994. Markov chains of lower Hessenberg-type, Working paper, University of Winnipeg.