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Sojourn time analysis of a two-phase queueing system with exhaustive batch-service and its vacation model
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MATHEMATICAL
www.ElsevierMathematics.com 63 l ONIIID
Mathematical
.I
and Computer
COMPUTER MODELLING
DIRECT.
SCICNCC
Modelling
38 (2003) 1283-1291 www.elsevier.com/locate/mcm
Sojourn Time Analysis of a Two-Phase Queueing System with Exhaustive Batch-Service and Its Vacation Model TSUYOSHI
KATAYAMA
AND KAORI
KOBAYASHI
Department
of Electronics and Informatics Faculty of Engineering Toyama Prefectural University Kosugi, Toyama 939-0398, Japan
Abstract-we
consider an M/G/l-type, two-phase queueing system, in which the two phases in series are attended alternatively and exhaustively by a moving single-server according to a batchservice in the first phase and an individual service in the second phase. We show that the two-phase queueing system reduces to a new type of single-vacation model with nonexhaustive service. Using a double transform for the joint distribution of the queue length in each phase and the remaining service time, we derive Laplace-Stieltjes transforms for the sojourn time in each phase and the total sojourn time in the system. Furthermore, we provide the moment formula of sojourn times and numerical examples of an approximate density function of the total sojourn time. @ 2003 Elsevier
Ltd. All rights reserved. Keywords-Two-phase model, Decomposition
service model, property, Semi-Markov
1.
Exhaustive process.
batch-service,
Sojourn
time,
Single-vacation
INTRODUCTION
We study a two-phase queueing system attended alternatively by a moving single-server, first analyzed by Krishna and Lee [l]. The server serves customers at the first phase according to the exhaustive batch-service, and then moves to the second phase. There, service is exhaustively done according to the individual service. (The details are described in the next section.) Practical examples of such a queueing model appear in the performance analysis of computer and communication systems and production systems as in [l-3]. On the other hand, similar single-server tandem queues with individual services for both stages have also been studied by Iravani et al. [4], Ivnitskiy and Kazatskiy [5], Katayama [6,7], Nair [8], and Taube-Netto [9]. We can consider the above two-phase service model with the exhaustive batch-service at the first phase as an approximate queueing model of the single-server tandem queues with exhaustive individual-services for both stages, first analyzed by Taube-Netto [9] by using the generating function (GF) approach. The motivation of this paper is due to a reduction of the two-phase service model to a new type of vacation model; besides, there are few articles discussed from the point of view concerning the decomposition property for single-server tandem queues as shown in [lo]. First, we derive a double transform for the joint distribution of the queue length in each phase and the remaining service time by using the theory of semi-Markov processes. Using the 0895-7177/03/$ - see front doi: 10.1016/S0895-7177(03)00341-8
matter
@ 2003
Elsevier
Ltd.
All rights
reserved.
Typeset
by A,+@-w
1284
T.
KATAYAMA
AND
K.
KOBAYASHI
double transform, we find the LSTs for the total sojourn time in the system and the sojourn time in each phase (Theorem l), which may be a new result based on the known generating function of the queue-length at batch-service starting-epochs. The results extend the work of Doshi [3] and Krishna and Lee [l]. Moreover, we present the decomposition form of the LST for the total sojourn time in the system (Corollary l), and provide explicit first and second moments formulas for the total sojourn time in the system and the sojourn time in each phase (Corollary 2). Using the results, we give numerical examples of an approximate density function of the total sojourn time. In Section 2, we describe the queueing model in detail. In Section 3, we derive a double transform for the stationary joint distribution of the queue length in each phase and the remaining service time. In Section 4, we analyse the total sojourn time in the system and the sojourn times in each phase. In Section 5, we give numerical examples. In Section 6, we summarize the paper.
2. MODEL
AND
NOTATION
We assume that the arrival process is Poisson at rate X. All customers go into the first queue Qi . On completion of the service at the first phase (Phase l), the customer is moved to the second queue Qz, and leaves the system after completing the service at the second phase (Phase 2). Customers in Qi and Qs are alternatively served by a moving single-server. Customers in Qi are served according to the exhaustive batch-service. The batch includes the customers present when the batch service starts as well as those that arrive during the batch service. The Phase-l service time Hr per an arbitrary batch is independent of the batch size. Customers in Q2 are individually served in the original order of arrival (FIFO rule). The Phase-2 service time Hz per an arbitrary customer is independent of Hi. The LST of the DF and the finite first three moments of H,, n = ‘1,2 are denoted by H:(s), h,, h!-,?, and hi3), respectively. Throughout, we will use the following notation: n = 1,2,
pn := Ah,,
Qn(x) := H;(X - Xx),
n = 1,2.
(1)
Additional notation will be introduced in Sections 3 and 4. For system stability, we assume that p2 is less than unity; see Remark 2.1. REMARK 2.1. The above two-phase queueing system can be reduced to an M/G/l vacation model with a virtual gate existing between Qi and Qz. The gate opens only when the server has returned from a vacation, and shuts just after all customers in Qi are moved to Qz. All customers in Qz are served individually in a service time Hz per customer. On completion of the service of all customers in Qz, the server takes a single vacation if there are one or more customers waiting in Qi. If Qi is empty, the server takes a single vacation just after a new arrival at &I, where the single vacation time is Hi. (Hence, Qr is not empty every time when the server returns from the vacation.) We will refer the modified single-vacation model with nonexhaustive service [lo] for our sojourn time analysis in Section 4.
3. RELATIONSHIP BETWEEN QUEUE-LENGTH GENERATING FUNCTIONS Let a nonnegative random variable xn(t), n = 1,2 denote the number of customers in Q,, n = 1,2 including the one being served at time t, and u(t) the location of a single server at time t. Then, we define a discrete-state, continuous parameter stochastic process {Y(t) : t 2 0) by y(t)
:=
{o(t),
xl(t)7
x2(t))r
(2)
where o(t) = n means that the server is serving at phase n, n = 1,2 at time t and a(t) = 0 that the server is idle because of xl(t) = x2(t) = 0. We denote the steady-state joint distribution
1285
Sojourn Time Analysis
of {Y(t)}
and the corresponding pn(i,j)
GF by
:=
pip?.
{a(t)
=
12, Xl(i)
i,j = 0,1,2 ,“‘,
7
i,
x2(t)
3
j)
9
n = 0, 1,2,
(3)
:= ~~;Pn(wY’, i=o j=o
Pn(X,Y)
1x1,IYI L 1,
n = 0,1,2.
(4
First, let t,, T = 1,2,. . . be the successive customer departure-epoch from Phase 1 or Phase 2. In addition, let t, - 0 (t,, + 0) be the epoch immediately before (after) the departure epoch t,, T = 1,2, . (We will use the same notation below.) We denote the steady-state joint distribution of an imbedded Markov chain {Y(t,.) : T = 1,2,. . .} by x,(i,j)
:= Jirnir P, {a(t,
- 0) = n, xr(t,
i,j=O,l,Z and the corresponding
).‘.,
+ 0) = i, xz(t, + 0) = j} , (5)
72= 1,2,
GF by
rI,(x,
y) := yy g 7r,(i,j)xiyj, i=o j=o
1x1,IYI L 1,
n = I,‘4
(6)
so that l&(0,1)
+ I&(1,1)
= 1.
(7)
In other words, 7rn(i,j) is the steady-state probability that there are i customers in Qr and j customers in Q2, when the server has completed a service for a customer in Q,, n = 1,2. Hence, note that rr(i,O) = 0 for any i 2 0, and ni(i,j) = 0 for i,j 2 1 because of the exhaustive batch-service in Phase 1. We denote by pzo the probability that the system is empty at a serverdeparture-epoch from Phase 2, i.e., psc := 7r2(O,O)/IIs(l, 0). Next, let t;, T = 1,2,. . , be the successive starting-epoch of a batch service. We denote the steady-state distribution of an imbedded Markov chain {Y (t;) : T = 1,2, . . . } by i=
p(i) := JLrnirP’ {a (t: + 0) = 1, x1 (tz + 0) = i, x2 (t: + 0) = 0)) and the corresponding
1,2,3...,
(8)
GF by P(x) := 2
1x1I 1,
p(i)x”,
(9)
i=l
where b(z) and p2a have been obtained by Doshi [3], which are given by (12) and (13) below. The explicit expressions for IIn(zr, y), n = 1,2 are then given by the following lemma. LEMMA
1. nl(o,Y)
n2(x7
')
=
1 + ,;I,“:+
=
I+
pl
Q~(Y)P(YL
p20
1 -
p2
-
p2
&2(x)
[QI(Y)~(Y)
+ PZO Y - &2(x)
- P(x)
- 1320(1-
x)1
>
(11)
where p(x)
=
fi
&I
[d%)]
- ~20 2 k=O
j=l
[I
- S(~)(X)]
fi
&I
[g(j)(x)]
,
W
j=l
fi Ql b’j’(O>l
i=l
p20
_I
= 1 +
5 k=O
[I
-
-
g(k+l)(o)]
-
;fi;
&I
(13) [g'j'(o)l
'
1286
T. KATAYAMA AND K. KOBAYASHI
and a sequence {g(“)(z)
: k = 0, 1, . . } is defined by O
k= 1,2,3 ,.... PROOF.
Considering a transition
from Y (t,.) to Y (&.+I), we obtain two functional
nl(o, Y)
= [~B(Y,
n2(z,y)
=
O)l QI(Y) + ~~~(O,O)YQI(Y)~
0) - ~2K4
[I~Iz(z,Y)
-
The GFs of the marginal distributions phase n, n = 1,2 are given by
relationships,
0)1&2(z); +
n2(2,
(15)
Y)Qz(~) ;.
~10%
(16)
of the number of customers at the service completion
at
(17)
P(Y>&I(Y> = $$$, 1,
P(z)= 1-
rJ2(3GO)
(18)
>
-P20
+p205,
n2(1,0)
where the last term p201t: in (18) corresponds to the case of the service for a new arrival during an idle period. Hence, from (15),(16) and (17),(18), we get (10) and (11). The coefficients on the right-hand sides of (10) and (11) can be obtained using the normalization condition (7). (The GF P(z) and ~20 are given by (3.9) and (3.11) in [3], respectively.) I Next, we define a stochastic vector process {YR(~) : t > 0) by
09)
YR(t) := {(T(t),Xl(t),X2(t),R,(t)(t)},
where h(t), n = 0, 1,2 denotes a remaining service time of Hn, n = 0, 1,2 of a customer in service at time t, where b(t) in a state {a(t) = 0) represents a remaining service time of a (virtual) service time I& with the LST
H,‘(s):=&. We denote the steady-state joint distribution pn(i,j;~)dr
:= LILIP,
of the vector process {YR(~) : t > 0) by
{o(t) = n, xl(t) i,j=O,l,Z
(20)
= i,
,“.,
x2(t)
=j,
Q-< R,(t)
L 7 +dT}, (21)
n = 0, 1,2.
We also let P,(z,y;s)
:= ~~P;(i,j;s)z’Y~, i=o j=o
where
1x1,Iy( 5 1,
n = 0, 1,2,
(22)
00 &(i, j; s) :=
s0
e-“p,,(i,j;
T) dT,
h(s)
> 0,
71= 0, 1,2.
Using Lemma 1, two stationary distributions for the vector processes {Y(t)) use in the next section have been determined completely as follows.
(23) and {YR(~)) to
LEMMA
1287
Time Analysis
Sojourn
2.
PI(x,0;s)= X(1 P2(T
y;s)= X(1
+ Pl
Hi(X
- P2 + P20)
Pl
(S -
+ P20
p1 + P20 -
-
Hi(S) - AS)
nl(o’z)’
H2*(X - Xx)- H,*(s) (s- x + Xs)H,*(X - AZ)*2(z3 y),
+ p1 - p2 + p2o)y
X(1
- AZ) X + XS)Hi(X
(24)
P2)P20
p”(07 O;s,= (p1+p2o)(s + A)’ E&c? Y)= Pn(G y;01, n = 0,1,2.
(25)
A relationship between II,(x, y), n = 1,2, and P,(z, y; s), n = 0,1,2 can be obtained by using the theory of semi-Markov processes as in [7]. (We omit the proof here, since the wellknown routine work is very lengthy.) Equation (25) is obtained by setting s = 0 in (24) from the definition of (4) and (22). I
PROOF.
4. SOJOURN In this section, we will total sojourn time in the sojourn time in phase n, Then we have 8 = 81 + THEOREM
TIME
ANALYSIS
find the LSTs of the DFs of the sojourn times in each phase and the system. Let @E(s), n = 1,2, and Q*(s) be the LSTs of the DFs of the 8,, n = 1,2 and the total sojourn time in the system 8, respectively. 02. Using Lemmas 1 and 2, we get the following results.
1. Q;(s) = poHi(s)
+ -xpoP PZOS
mw,wl
,
(26)
Q;(s)- po H2*(s) P - fml 7 P20 1- q?(s) Q*(s)
= 5
S
Hz*(s)
[Pzos
~20 s - X + AH,*(s) = n2 (1 - s/h
(27) X (1 - H;(s)}
+
I’ (1
x
>I
1 - s/N
(29)
’
n,(l,l)
(28)
where p. := PO((), 0) = (l - l@)mo ) Pl 444
:=
&I [H;(s)]
I’
(30)
+ P20 [H;(s)]
.
We consider the modified single-vacation model with nonexhaustive service and a gate existing between Qi and Q2 described in Remark 2.1. That is, the state (o(t) = 1) represents a state that the server is in vacation time HI. The state probability just before arrival of an arbitrary tagged customer is equal to the stationary state probability defined in (21) because of the PASTA (Poisson arrivals see time averages) property. Here, recall that in pA(i, j; s) defined in (23), the number of i (j) customers includes the one in service. Therefore, it follows from the vacation rule and the FIFO rule at Phase 2 that PROOF.
Q;(S) = poH;(s)
+ f&:(&O;
s) + 5 i=o
i=l
= poHf(s) Q;(s) =poH;(s)
+ Pi&O;
s) +
s)H,*(s)‘-lHi(s)
(31)
j=l
Hi(s) H;(s)
(32)
-J’2(LH;(s);s),
+ ~p;(i,O;O)H;(s)“+’ i=l
FpG(i,j;
+ ~~~~~;(~,~;o)H~(s)~+’ i=O j=l
(33)
1288
T. KATAYAMA
=
e*(s)
ff,*(s)
[PO + J3 l%(s),
= poH;(s)H;(s)
AND K.
KOBAYASHI
0; 0) + p2 (Hz*(s),
+ glE(i,
(34)
1; O)] ,
0; s)H;(s)i+l
i=l
+ T: ~;p;(i, j;s)H;(s)H;(s)“+j +lJj=l = Pofc(S)fG(S)
+
+ H;(s)P2
(35)
~2*(S)Pl(~2*(~),
m(s),
qw;
0; s>
(36)
8) ,
where the last two terms on the right-hand side of (33) correspond to the case that tagged customer has arrived at Qs, there are i customers waiting before the tagged After some algebraic manipulations of (32), (34), and (36) by using Lemmas 1 and obtain (26), (27), and (28)) respectively. The expression of (29) is known as the distributional form of Little’s law).
when the customer. 2, we thus LLD (the I REMARK 4.1. The LLD holds for most of M/G/I-type queues with FIFO rule, however, which does not hold for the following few models because of the correlation of the arrival counting process and the total sojourn time; the N-policy vacation model [ll] and the single-server tandem queues with individual services for both stages described in Section 1 [7].
The LST 0*( s) in Theorem 1 is also expressed by the decomposition form, applying the Fuhrmann-Cooper decomposition to the modified single-vacation model with nonexhaustive service. Two decomposition forms and the probabilistic interpretation are given as follows. 1.
COROLLARY o*(s)
=
(1 - P2wwS)
1 - P2 -
su,o>
+
p
4(LO)
s - x + AH;(s)
1 -
1_ s (
p2
1 - T(s) >
I
(37)
and
where
1 1 -&Jo
nz(G
:= ~
l&(z)
1
0)
n,o
-P20
and PI3
:=
Pl(L
(1 -Pzo)
PO@,
PI := 1 - Ps =
0)
=
0) + su,
P2oQ
&v(x)
’
0)
Pl(l
:=
x&l(x),
(39)
’
(40)
-P20)
p1+p20
+ Pl>
p1+?320
.
The e*(s) can be interpreted as a product of the LST for the virtual workload at an arbitrary time and Hz(s), where the virtual workload is the sum of the workload in the system and the remaining vacation time. We then have a decomposition expression for O*(s), (37)) using Theorem 3.3 in [12] for M/G/l queue with general server vacations, since our vacation model satisfies Assumptions (i)-(v), (iii’), and (iii”) in [12], and an additional condition of the independence of the workload present at the beginning of a vacation and the vacation period. The second factor on the right-hand side of (37) represents the LST for the workload at an arbitrary time during nonbusy periods (= idle periods + vacations corresponding to the states {o(t) = 0 and l}, respectively). The other decomposition form (38) can be derived as
PROOF.
Q*P
-
Xx)
=
l32(x, n2(&
x) l)
=
(1 - ~210
&2(x)
- ~)&2(2) -x 1 -
’
pB=2(x)
&I(X)
(1 - x)Q;(l)
(41) 1 -
&v(x)
+ pr (1 - x)Q;(l)
1 ’
Sojourn
Time
1289
Analysis
where II,(z) is the GF of the number of customers left behind when a vacation starts in the case that the system is nonempty, and the first term in the square bracket in (41) can be obtained by using the three-way decomposition for nonexhaustive service M/G/l vacation models; see equation (6.16) in [lo]. The second term corresponding to the case that the system becomes empty can directly be derived from the general result of the decomposition for an exhaustiveservice M/G/l vacation model, where Q,,(X) is the GF of the number of customers present at 1 the beginning of a busy period (= at the end of a vacation period); see Section 4.2 in [lo]. 4.2. Note that the decomposition theorem gives no closed expression for O*(s) as in (37) and (38), since the equations contain unknown probabilities, PB, PI, and PI (1,O). That is, we require necessarily Lemma 2 in the last section to derive a final equation from (37) and (38). We can obtain a final expression without unknown probabilities for O*(s) from (29) using only Lemma 1. Finally, four expressions for e*(s), i.e., (28), (29), (37), and (38) have been provided. It is seen from the above argument that the probabilistic structure of our vacation model and the relationship between the two-phase queueing system with exhaustive batch-service and our vacation model. REMARK
Lastly, we provide the mean sojourn time formulas and the second moment formula. COROLLARY
2. PlP2
1
+ P20 hl
E(Q1)
=
(1 -
E(Q2)
=
(1 -
&(Pl
+ P20)
P2hP2
+ P20)
P$)(Pl
hl
w
+
PZ)(Pl
+pzo)
l
20
l -
Ah?‘, Pi>
hz &c2)
p2
+
+ P20)
2 (1
l
-
P2)(Pl
Ah?),
h1 + h2 +
+ P20)
5%
(42)
PZ>
1
+ P20
PlP2
E(Q) = (1 -
+
XhC2) +
‘41 + pa)(p1
+ P20) P2
+
+
Ah?) +
fP20)
Xhy) ,
1 w
-
P2)
(43)
and 2hP2
E P2) 1
+-
REMARK
4.3.
l+ E(8)
+ P20
+ P20)
(1 +
PlP2
-
Pi)
(p1 +p20)(1 - pz) (1 - pi) P2 +
PI
+
= 1
(1 - p21;;1 - p;,
(1 +
I
P2)P20
rw2(1+
+
2P2)
1-p;.
Ah?)
hg) + 3h
h1h2 l
Ah?) + 3(1
+Pzo)
(44
1
h(2)
+ -
P2)
Ah?) -E(O). 1 - p2
I
is identical with E(Z) in Section 3.3 in [3]; however, a clerical error is seen
in (3.20) in [3].
5. NUMERICAL
EXAMPLES
Let 0(t) be the PDF (probability density function) of the total sojourn time in the system 0. Using Corollary 2, an approximate PDF e(t) can be calculated by applying the two moments matching method using E(8) and E(e2) to a gamma distribution in the case e(O+) = 0 and, e.g., a hyperexponential distribution in the case 0(0+) > 0, where unknown parameters appearing in such fitting functions can be determined by using E(8), E(e2), and (46) given below. Assuming that Hz(t) is a k-phase Erlangian distribution with the mean value l/p, we then have 8(0+) = JimmsQ*(s)
= ssz X
1 - p2 1 - (X - XH,*(s))/s 1 - p2 - Pl(l,O) 1 -
p2
[ b 1k
(45)
s s+kp
+ Pl(l,O) 1 -
p P2
l-s (
l-K(s) x >
shl
1 ;
1290
T. KATAYAMA
AND
K. KOBAYASHI
that is,
qo+>= o(1 -
~2
-
for k = 1,
P1(1,O))p,
for k 2 2.
7
(Here note that J*(s) := P(l - s/X) rep resents the LST for the virtual workload seen at the beginning of the first-phase service.) Figure 1 shows some numerical examples of the PDF 0(t) for the number of phases k = 1, 10, and 100, where it is assumed that Hi(t) is an exponential distribution function with the mean hi = 1, and Hz(t) is a k-phase Erlangian distribution with the mean h2 = 5. 0.03
0.025
3 5 .-
f
0.02
0.015
if B 1 a
0.01
0.005
0
0
20
40
60
50
100
120
140
160
,
Figure
1. Probability
density
function:
0(t).
6. CONCLUSIONS For the two-phase queueing system with exhaustive batch-service and individual service, which is a more realistic model of a single router, we have derived the double transform of the joint distribution of queue length in each phase’and the remaining service time. Using the result, we have derived the LSTs and the first and second moments of the sojourn time in each phase and the total sojourn time in the system. We have also found the relationship between the two-phase queueing system and the modified single-vacation model, and have presented the probabilistic structure for the two-phase queueing system using the Fuhrmann-Cooper decomposition. It should be noted for theoretical analysis that the two-stage queueing system with individual services for both stages as in [4,9] similar to our vacation model can be reduced to an M/G/l queue with vacations, however, in which the vacation time is strongly correlated with the past service session.
REFERENCES 1. C.M. Krishna and Y.-H. Lee, A study of two-phase service, Oper. Res. Letters 9, 91-97, (1990). 2. D. De&money Selvam and V. Sivasankaran, A two-phase queueing system with server vacations, Oper. Res. Letters 15, 163-168, (1994). 3. B.T. Doshi, Analysis of a two phase queueing system with general service times, Oper. Res. Letters 10, 265-272, (1991). 4. S.M.R. Iravani, M.J.M. Posner and J.A. Buzacott, A two-stage tandem queue attended by a moving server with holding and switching costs, Queueing Systems 26, 203-228, (1997). 5. V.A. Ivnitskiy and V.P. Kazatskiy, On a two-phase queueing system, Engineering Cybernetics 10, 1050-1057, (1972).
Sojourn
Time
Analysis
1291
6. T. Katayama, Performance analysis and optimization of a cyclic-service tandem queueing system with multiclass customers, Computers Math. Applic. 24 (l/2), 25-34, (1992). 7. T. Katayama, A note on sojourn time analysis of a two-stage queueing system, Stochastic Models 15, 379-394, (1999). 8. S.S. Nair, A single server tandem queue, J. Appl. Prob. 8, 95-109, (1971). 9. M. Taube-Netto, Two queues in tandem attended by a single server, Oper. Res. 25, 140-147, (1997). 10. B.T. Doshi, Queueing systems with vacations-A survey, Queueing Systems 1, 29-66, (1986). Volume 1, Vacatzon and Priority 11. H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Systems, Part 1, Elsevier Science, North-Holland, (1991). 12. M. Miyazawa, Decomposition formula for a single server queue with vacations: A unified approach by the rate conservation law, Stochastic Models 10, 389-413, (1994). 13. D. Towsley, A study of a queueing system with three-phase service, Oper. Res. Letters 13, 189-195, (1993).
MATHEMATICAL
www.ElsevierMathematics.com 63 l ONIIID
Mathematical
.I
and Computer
COMPUTER MODELLING
DIRECT.
SCICNCC
Modelling
38 (2003) 1283-1291 www.elsevier.com/locate/mcm
Sojourn Time Analysis of a Two-Phase Queueing System with Exhaustive Batch-Service and Its Vacation Model TSUYOSHI
KATAYAMA
AND KAORI
KOBAYASHI
Department
of Electronics and Informatics Faculty of Engineering Toyama Prefectural University Kosugi, Toyama 939-0398, Japan
Abstract-we
consider an M/G/l-type, two-phase queueing system, in which the two phases in series are attended alternatively and exhaustively by a moving single-server according to a batchservice in the first phase and an individual service in the second phase. We show that the two-phase queueing system reduces to a new type of single-vacation model with nonexhaustive service. Using a double transform for the joint distribution of the queue length in each phase and the remaining service time, we derive Laplace-Stieltjes transforms for the sojourn time in each phase and the total sojourn time in the system. Furthermore, we provide the moment formula of sojourn times and numerical examples of an approximate density function of the total sojourn time. @ 2003 Elsevier
Ltd. All rights reserved. Keywords-Two-phase model, Decomposition
service model, property, Semi-Markov
1.
Exhaustive process.
batch-service,
Sojourn
time,
Single-vacation
INTRODUCTION
We study a two-phase queueing system attended alternatively by a moving single-server, first analyzed by Krishna and Lee [l]. The server serves customers at the first phase according to the exhaustive batch-service, and then moves to the second phase. There, service is exhaustively done according to the individual service. (The details are described in the next section.) Practical examples of such a queueing model appear in the performance analysis of computer and communication systems and production systems as in [l-3]. On the other hand, similar single-server tandem queues with individual services for both stages have also been studied by Iravani et al. [4], Ivnitskiy and Kazatskiy [5], Katayama [6,7], Nair [8], and Taube-Netto [9]. We can consider the above two-phase service model with the exhaustive batch-service at the first phase as an approximate queueing model of the single-server tandem queues with exhaustive individual-services for both stages, first analyzed by Taube-Netto [9] by using the generating function (GF) approach. The motivation of this paper is due to a reduction of the two-phase service model to a new type of vacation model; besides, there are few articles discussed from the point of view concerning the decomposition property for single-server tandem queues as shown in [lo]. First, we derive a double transform for the joint distribution of the queue length in each phase and the remaining service time by using the theory of semi-Markov processes. Using the 0895-7177/03/$ - see front doi: 10.1016/S0895-7177(03)00341-8
matter
@ 2003
Elsevier
Ltd.
All rights
reserved.
Typeset
by A,+@-w
1284
T.
KATAYAMA
AND
K.
KOBAYASHI
double transform, we find the LSTs for the total sojourn time in the system and the sojourn time in each phase (Theorem l), which may be a new result based on the known generating function of the queue-length at batch-service starting-epochs. The results extend the work of Doshi [3] and Krishna and Lee [l]. Moreover, we present the decomposition form of the LST for the total sojourn time in the system (Corollary l), and provide explicit first and second moments formulas for the total sojourn time in the system and the sojourn time in each phase (Corollary 2). Using the results, we give numerical examples of an approximate density function of the total sojourn time. In Section 2, we describe the queueing model in detail. In Section 3, we derive a double transform for the stationary joint distribution of the queue length in each phase and the remaining service time. In Section 4, we analyse the total sojourn time in the system and the sojourn times in each phase. In Section 5, we give numerical examples. In Section 6, we summarize the paper.
2. MODEL
AND
NOTATION
We assume that the arrival process is Poisson at rate X. All customers go into the first queue Qi . On completion of the service at the first phase (Phase l), the customer is moved to the second queue Qz, and leaves the system after completing the service at the second phase (Phase 2). Customers in Qi and Qs are alternatively served by a moving single-server. Customers in Qi are served according to the exhaustive batch-service. The batch includes the customers present when the batch service starts as well as those that arrive during the batch service. The Phase-l service time Hr per an arbitrary batch is independent of the batch size. Customers in Q2 are individually served in the original order of arrival (FIFO rule). The Phase-2 service time Hz per an arbitrary customer is independent of Hi. The LST of the DF and the finite first three moments of H,, n = ‘1,2 are denoted by H:(s), h,, h!-,?, and hi3), respectively. Throughout, we will use the following notation: n = 1,2,
pn := Ah,,
Qn(x) := H;(X - Xx),
n = 1,2.
(1)
Additional notation will be introduced in Sections 3 and 4. For system stability, we assume that p2 is less than unity; see Remark 2.1. REMARK 2.1. The above two-phase queueing system can be reduced to an M/G/l vacation model with a virtual gate existing between Qi and Qz. The gate opens only when the server has returned from a vacation, and shuts just after all customers in Qi are moved to Qz. All customers in Qz are served individually in a service time Hz per customer. On completion of the service of all customers in Qz, the server takes a single vacation if there are one or more customers waiting in Qi. If Qi is empty, the server takes a single vacation just after a new arrival at &I, where the single vacation time is Hi. (Hence, Qr is not empty every time when the server returns from the vacation.) We will refer the modified single-vacation model with nonexhaustive service [lo] for our sojourn time analysis in Section 4.
3. RELATIONSHIP BETWEEN QUEUE-LENGTH GENERATING FUNCTIONS Let a nonnegative random variable xn(t), n = 1,2 denote the number of customers in Q,, n = 1,2 including the one being served at time t, and u(t) the location of a single server at time t. Then, we define a discrete-state, continuous parameter stochastic process {Y(t) : t 2 0) by y(t)
:=
{o(t),
xl(t)7
x2(t))r
(2)
where o(t) = n means that the server is serving at phase n, n = 1,2 at time t and a(t) = 0 that the server is idle because of xl(t) = x2(t) = 0. We denote the steady-state joint distribution
1285
Sojourn Time Analysis
of {Y(t)}
and the corresponding pn(i,j)
GF by
:=
pip?.
{a(t)
=
12, Xl(i)
i,j = 0,1,2 ,“‘,
7
i,
x2(t)
3
j)
9
n = 0, 1,2,
(3)
:= ~~;Pn(wY’, i=o j=o
Pn(X,Y)
1x1,IYI L 1,
n = 0,1,2.
(4
First, let t,, T = 1,2,. . . be the successive customer departure-epoch from Phase 1 or Phase 2. In addition, let t, - 0 (t,, + 0) be the epoch immediately before (after) the departure epoch t,, T = 1,2, . (We will use the same notation below.) We denote the steady-state joint distribution of an imbedded Markov chain {Y(t,.) : T = 1,2,. . .} by x,(i,j)
:= Jirnir P, {a(t,
- 0) = n, xr(t,
i,j=O,l,Z and the corresponding
).‘.,
+ 0) = i, xz(t, + 0) = j} , (5)
72= 1,2,
GF by
rI,(x,
y) := yy g 7r,(i,j)xiyj, i=o j=o
1x1,IYI L 1,
n = I,‘4
(6)
so that l&(0,1)
+ I&(1,1)
= 1.
(7)
In other words, 7rn(i,j) is the steady-state probability that there are i customers in Qr and j customers in Q2, when the server has completed a service for a customer in Q,, n = 1,2. Hence, note that rr(i,O) = 0 for any i 2 0, and ni(i,j) = 0 for i,j 2 1 because of the exhaustive batch-service in Phase 1. We denote by pzo the probability that the system is empty at a serverdeparture-epoch from Phase 2, i.e., psc := 7r2(O,O)/IIs(l, 0). Next, let t;, T = 1,2,. . , be the successive starting-epoch of a batch service. We denote the steady-state distribution of an imbedded Markov chain {Y (t;) : T = 1,2, . . . } by i=
p(i) := JLrnirP’ {a (t: + 0) = 1, x1 (tz + 0) = i, x2 (t: + 0) = 0)) and the corresponding
1,2,3...,
(8)
GF by P(x) := 2
1x1I 1,
p(i)x”,
(9)
i=l
where b(z) and p2a have been obtained by Doshi [3], which are given by (12) and (13) below. The explicit expressions for IIn(zr, y), n = 1,2 are then given by the following lemma. LEMMA
1. nl(o,Y)
n2(x7
')
=
1 + ,;I,“:+
=
I+
pl
Q~(Y)P(YL
p20
1 -
p2
-
p2
&2(x)
[QI(Y)~(Y)
+ PZO Y - &2(x)
- P(x)
- 1320(1-
x)1
>
(11)
where p(x)
=
fi
&I
[d%)]
- ~20 2 k=O
j=l
[I
- S(~)(X)]
fi
&I
[g(j)(x)]
,
W
j=l
fi Ql b’j’(O>l
i=l
p20
_I
= 1 +
5 k=O
[I
-
-
g(k+l)(o)]
-
;fi;
&I
(13) [g'j'(o)l
'
1286
T. KATAYAMA AND K. KOBAYASHI
and a sequence {g(“)(z)
: k = 0, 1, . . } is defined by O
k= 1,2,3 ,.... PROOF.
Considering a transition
from Y (t,.) to Y (&.+I), we obtain two functional
nl(o, Y)
= [~B(Y,
n2(z,y)
=
O)l QI(Y) + ~~~(O,O)YQI(Y)~
0) - ~2K4
[I~Iz(z,Y)
-
The GFs of the marginal distributions phase n, n = 1,2 are given by
relationships,
0)1&2(z); +
n2(2,
(15)
Y)Qz(~) ;.
~10%
(16)
of the number of customers at the service completion
at
(17)
P(Y>&I(Y> = $$$, 1,
P(z)= 1-
rJ2(3GO)
(18)
>
-P20
+p205,
n2(1,0)
where the last term p201t: in (18) corresponds to the case of the service for a new arrival during an idle period. Hence, from (15),(16) and (17),(18), we get (10) and (11). The coefficients on the right-hand sides of (10) and (11) can be obtained using the normalization condition (7). (The GF P(z) and ~20 are given by (3.9) and (3.11) in [3], respectively.) I Next, we define a stochastic vector process {YR(~) : t > 0) by
09)
YR(t) := {(T(t),Xl(t),X2(t),R,(t)(t)},
where h(t), n = 0, 1,2 denotes a remaining service time of Hn, n = 0, 1,2 of a customer in service at time t, where b(t) in a state {a(t) = 0) represents a remaining service time of a (virtual) service time I& with the LST
H,‘(s):=&. We denote the steady-state joint distribution pn(i,j;~)dr
:= LILIP,
of the vector process {YR(~) : t > 0) by
{o(t) = n, xl(t) i,j=O,l,Z
(20)
= i,
,“.,
x2(t)
=j,
Q-< R,(t)
L 7 +dT}, (21)
n = 0, 1,2.
We also let P,(z,y;s)
:= ~~P;(i,j;s)z’Y~, i=o j=o
where
1x1,Iy( 5 1,
n = 0, 1,2,
(22)
00 &(i, j; s) :=
s0
e-“p,,(i,j;
T) dT,
h(s)
> 0,
71= 0, 1,2.
Using Lemma 1, two stationary distributions for the vector processes {Y(t)) use in the next section have been determined completely as follows.
(23) and {YR(~)) to
LEMMA
1287
Time Analysis
Sojourn
2.
PI(x,0;s)= X(1 P2(T
y;s)= X(1
+ Pl
Hi(X
- P2 + P20)
Pl
(S -
+ P20
p1 + P20 -
-
Hi(S) - AS)
nl(o’z)’
H2*(X - Xx)- H,*(s) (s- x + Xs)H,*(X - AZ)*2(z3 y),
+ p1 - p2 + p2o)y
X(1
- AZ) X + XS)Hi(X
(24)
P2)P20
p”(07 O;s,= (p1+p2o)(s + A)’ E&c? Y)= Pn(G y;01, n = 0,1,2.
(25)
A relationship between II,(x, y), n = 1,2, and P,(z, y; s), n = 0,1,2 can be obtained by using the theory of semi-Markov processes as in [7]. (We omit the proof here, since the wellknown routine work is very lengthy.) Equation (25) is obtained by setting s = 0 in (24) from the definition of (4) and (22). I
PROOF.
4. SOJOURN In this section, we will total sojourn time in the sojourn time in phase n, Then we have 8 = 81 + THEOREM
TIME
ANALYSIS
find the LSTs of the DFs of the sojourn times in each phase and the system. Let @E(s), n = 1,2, and Q*(s) be the LSTs of the DFs of the 8,, n = 1,2 and the total sojourn time in the system 8, respectively. 02. Using Lemmas 1 and 2, we get the following results.
1. Q;(s) = poHi(s)
+ -xpoP PZOS
mw,wl
,
(26)
Q;(s)- po H2*(s) P - fml 7 P20 1- q?(s) Q*(s)
= 5
S
Hz*(s)
[Pzos
~20 s - X + AH,*(s) = n2 (1 - s/h
(27) X (1 - H;(s)}
+
I’ (1
x
>I
1 - s/N
(29)
’
n,(l,l)
(28)
where p. := PO((), 0) = (l - l@)mo ) Pl 444
:=
&I [H;(s)]
I’
(30)
+ P20 [H;(s)]
.
We consider the modified single-vacation model with nonexhaustive service and a gate existing between Qi and Q2 described in Remark 2.1. That is, the state (o(t) = 1) represents a state that the server is in vacation time HI. The state probability just before arrival of an arbitrary tagged customer is equal to the stationary state probability defined in (21) because of the PASTA (Poisson arrivals see time averages) property. Here, recall that in pA(i, j; s) defined in (23), the number of i (j) customers includes the one in service. Therefore, it follows from the vacation rule and the FIFO rule at Phase 2 that PROOF.
Q;(S) = poH;(s)
+ f&:(&O;
s) + 5 i=o
i=l
= poHf(s) Q;(s) =poH;(s)
+ Pi&O;
s) +
s)H,*(s)‘-lHi(s)
(31)
j=l
Hi(s) H;(s)
(32)
-J’2(LH;(s);s),
+ ~p;(i,O;O)H;(s)“+’ i=l
FpG(i,j;
+ ~~~~~;(~,~;o)H~(s)~+’ i=O j=l
(33)
1288
T. KATAYAMA
=
e*(s)
ff,*(s)
[PO + J3 l%(s),
= poH;(s)H;(s)
AND K.
KOBAYASHI
0; 0) + p2 (Hz*(s),
+ glE(i,
(34)
1; O)] ,
0; s)H;(s)i+l
i=l
+ T: ~;p;(i, j;s)H;(s)H;(s)“+j +lJj=l = Pofc(S)fG(S)
+
+ H;(s)P2
(35)
~2*(S)Pl(~2*(~),
m(s),
qw;
0; s>
(36)
8) ,
where the last two terms on the right-hand side of (33) correspond to the case that tagged customer has arrived at Qs, there are i customers waiting before the tagged After some algebraic manipulations of (32), (34), and (36) by using Lemmas 1 and obtain (26), (27), and (28)) respectively. The expression of (29) is known as the distributional form of Little’s law).
when the customer. 2, we thus LLD (the I REMARK 4.1. The LLD holds for most of M/G/I-type queues with FIFO rule, however, which does not hold for the following few models because of the correlation of the arrival counting process and the total sojourn time; the N-policy vacation model [ll] and the single-server tandem queues with individual services for both stages described in Section 1 [7].
The LST 0*( s) in Theorem 1 is also expressed by the decomposition form, applying the Fuhrmann-Cooper decomposition to the modified single-vacation model with nonexhaustive service. Two decomposition forms and the probabilistic interpretation are given as follows. 1.
COROLLARY o*(s)
=
(1 - P2wwS)
1 - P2 -
su,o>
+
p
4(LO)
s - x + AH;(s)
1 -
1_ s (
p2
1 - T(s) >
I
(37)
and
where
1 1 -&Jo
nz(G
:= ~
l&(z)
1
0)
n,o
-P20
and PI3
:=
Pl(L
(1 -Pzo)
PO@,
PI := 1 - Ps =
0)
=
0) + su,
P2oQ
&v(x)
’
0)
Pl(l
:=
x&l(x),
(39)
’
(40)
-P20)
p1+p20
+ Pl>
p1+?320
.
The e*(s) can be interpreted as a product of the LST for the virtual workload at an arbitrary time and Hz(s), where the virtual workload is the sum of the workload in the system and the remaining vacation time. We then have a decomposition expression for O*(s), (37)) using Theorem 3.3 in [12] for M/G/l queue with general server vacations, since our vacation model satisfies Assumptions (i)-(v), (iii’), and (iii”) in [12], and an additional condition of the independence of the workload present at the beginning of a vacation and the vacation period. The second factor on the right-hand side of (37) represents the LST for the workload at an arbitrary time during nonbusy periods (= idle periods + vacations corresponding to the states {o(t) = 0 and l}, respectively). The other decomposition form (38) can be derived as
PROOF.
Q*P
-
Xx)
=
l32(x, n2(&
x) l)
=
(1 - ~210
&2(x)
- ~)&2(2) -x 1 -
’
pB=2(x)
&I(X)
(1 - x)Q;(l)
(41) 1 -
&v(x)
+ pr (1 - x)Q;(l)
1 ’
Sojourn
Time
1289
Analysis
where II,(z) is the GF of the number of customers left behind when a vacation starts in the case that the system is nonempty, and the first term in the square bracket in (41) can be obtained by using the three-way decomposition for nonexhaustive service M/G/l vacation models; see equation (6.16) in [lo]. The second term corresponding to the case that the system becomes empty can directly be derived from the general result of the decomposition for an exhaustiveservice M/G/l vacation model, where Q,,(X) is the GF of the number of customers present at 1 the beginning of a busy period (= at the end of a vacation period); see Section 4.2 in [lo]. 4.2. Note that the decomposition theorem gives no closed expression for O*(s) as in (37) and (38), since the equations contain unknown probabilities, PB, PI, and PI (1,O). That is, we require necessarily Lemma 2 in the last section to derive a final equation from (37) and (38). We can obtain a final expression without unknown probabilities for O*(s) from (29) using only Lemma 1. Finally, four expressions for e*(s), i.e., (28), (29), (37), and (38) have been provided. It is seen from the above argument that the probabilistic structure of our vacation model and the relationship between the two-phase queueing system with exhaustive batch-service and our vacation model. REMARK
Lastly, we provide the mean sojourn time formulas and the second moment formula. COROLLARY
2. PlP2
1
+ P20 hl
E(Q1)
=
(1 -
E(Q2)
=
(1 -
&(Pl
+ P20)
P2hP2
+ P20)
P$)(Pl
hl
w
+
PZ)(Pl
+pzo)
l
20
l -
Ah?‘, Pi>
hz &c2)
p2
+
+ P20)
2 (1
l
-
P2)(Pl
Ah?),
h1 + h2 +
+ P20)
5%
(42)
PZ>
1
+ P20
PlP2
E(Q) = (1 -
+
XhC2) +
‘41 + pa)(p1
+ P20) P2
+
+
Ah?) +
fP20)
Xhy) ,
1 w
-
P2)
(43)
and 2hP2
E P2) 1
+-
REMARK
4.3.
l+ E(8)
+ P20
+ P20)
(1 +
PlP2
-
Pi)
(p1 +p20)(1 - pz) (1 - pi) P2 +
PI
+
= 1
(1 - p21;;1 - p;,
(1 +
I
P2)P20
rw2(1+
+
2P2)
1-p;.
Ah?)
hg) + 3h
h1h2 l
Ah?) + 3(1
+Pzo)
(44
1
h(2)
+ -
P2)
Ah?) -E(O). 1 - p2
I
is identical with E(Z) in Section 3.3 in [3]; however, a clerical error is seen
in (3.20) in [3].
5. NUMERICAL
EXAMPLES
Let 0(t) be the PDF (probability density function) of the total sojourn time in the system 0. Using Corollary 2, an approximate PDF e(t) can be calculated by applying the two moments matching method using E(8) and E(e2) to a gamma distribution in the case e(O+) = 0 and, e.g., a hyperexponential distribution in the case 0(0+) > 0, where unknown parameters appearing in such fitting functions can be determined by using E(8), E(e2), and (46) given below. Assuming that Hz(t) is a k-phase Erlangian distribution with the mean value l/p, we then have 8(0+) = JimmsQ*(s)
= ssz X
1 - p2 1 - (X - XH,*(s))/s 1 - p2 - Pl(l,O) 1 -
p2
[ b 1k
(45)
s s+kp
+ Pl(l,O) 1 -
p P2
l-s (
l-K(s) x >
shl
1 ;
1290
T. KATAYAMA
AND
K. KOBAYASHI
that is,
qo+>= o(1 -
~2
-
for k = 1,
P1(1,O))p,
for k 2 2.
7
(Here note that J*(s) := P(l - s/X) rep resents the LST for the virtual workload seen at the beginning of the first-phase service.) Figure 1 shows some numerical examples of the PDF 0(t) for the number of phases k = 1, 10, and 100, where it is assumed that Hi(t) is an exponential distribution function with the mean hi = 1, and Hz(t) is a k-phase Erlangian distribution with the mean h2 = 5. 0.03
0.025
3 5 .-
f
0.02
0.015
if B 1 a
0.01
0.005
0
0
20
40
60
50
100
120
140
160
,
Figure
1. Probability
density
function:
0(t).
6. CONCLUSIONS For the two-phase queueing system with exhaustive batch-service and individual service, which is a more realistic model of a single router, we have derived the double transform of the joint distribution of queue length in each phase’and the remaining service time. Using the result, we have derived the LSTs and the first and second moments of the sojourn time in each phase and the total sojourn time in the system. We have also found the relationship between the two-phase queueing system and the modified single-vacation model, and have presented the probabilistic structure for the two-phase queueing system using the Fuhrmann-Cooper decomposition. It should be noted for theoretical analysis that the two-stage queueing system with individual services for both stages as in [4,9] similar to our vacation model can be reduced to an M/G/l queue with vacations, however, in which the vacation time is strongly correlated with the past service session.
REFERENCES 1. C.M. Krishna and Y.-H. Lee, A study of two-phase service, Oper. Res. Letters 9, 91-97, (1990). 2. D. De&money Selvam and V. Sivasankaran, A two-phase queueing system with server vacations, Oper. Res. Letters 15, 163-168, (1994). 3. B.T. Doshi, Analysis of a two phase queueing system with general service times, Oper. Res. Letters 10, 265-272, (1991). 4. S.M.R. Iravani, M.J.M. Posner and J.A. Buzacott, A two-stage tandem queue attended by a moving server with holding and switching costs, Queueing Systems 26, 203-228, (1997). 5. V.A. Ivnitskiy and V.P. Kazatskiy, On a two-phase queueing system, Engineering Cybernetics 10, 1050-1057, (1972).
Sojourn
Time
Analysis
1291
6. T. Katayama, Performance analysis and optimization of a cyclic-service tandem queueing system with multiclass customers, Computers Math. Applic. 24 (l/2), 25-34, (1992). 7. T. Katayama, A note on sojourn time analysis of a two-stage queueing system, Stochastic Models 15, 379-394, (1999). 8. S.S. Nair, A single server tandem queue, J. Appl. Prob. 8, 95-109, (1971). 9. M. Taube-Netto, Two queues in tandem attended by a single server, Oper. Res. 25, 140-147, (1997). 10. B.T. Doshi, Queueing systems with vacations-A survey, Queueing Systems 1, 29-66, (1986). Volume 1, Vacatzon and Priority 11. H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Systems, Part 1, Elsevier Science, North-Holland, (1991). 12. M. Miyazawa, Decomposition formula for a single server queue with vacations: A unified approach by the rate conservation law, Stochastic Models 10, 389-413, (1994). 13. D. Towsley, A study of a queueing system with three-phase service, Oper. Res. Letters 13, 189-195, (1993).