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Sojourn time analysis for a cyclic-service tandem queueing model with general decrementing service
Mathl. Comput.
Pergamon
Modelling
Vol. 22, No. 19-12,
pp. 131-139,
1995
Copyright@1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 08957177/95 $9.50 + 0.99
0895-7177(95)00189-l
Sojourn Time Analysis for a Cyclic-Service Tandem Queueing Model with General Decrementing Service T.
KATAYAMA AND K. KOBAYASHI Faculty of Engineering Toyama Prefectural University Kosugi-Machi, Toyama 939-03, Japan
Abstract-A sojourn time analysis is provided for a cyclic-service tandem queue with general decrementing service which operates as follows: starting once a service of queue 1 in the first stage, a single server continues serving messages in queue 1 until either queue 1 becomes empty, or the number of messages decreases to k lees than that found upon the server’s last arrival at queue 1, whichever occurs first, where 1 5 k < 00. After service completion in queue 1, the server switches over to queue 2 in the second stage and serves all messages in queue 2 until it becomes empty. It is assumed that an arrival stream is Poissonian, message service times at eech stage are generally distributed, and switch-over times are zero. This paper analyzes joint queue-length distributions and message sojourn time distributions. Keywords-General tion.
decrementingservice,Cyclic-servicetandemqueue, Sojourn time distribu-
1. INTRODUCTION There are several practical examples of tandem queueing models served by a single server in the performance evaluation of computer operating systems and call processing in switching systems in telecommunication networks [1,2]. This paper considers such a tandem queueing model with general decrementing service of a parameter k which operates as follows [3]. For the moment, suppose that messages in the first stage are served. A single server continues serving messages in queue 1 until either queue 1 becomes empty, or the number of messages decreases to k less than that found upon the server’s last arrival at queue 1, whichever occurs first. We also call this service discipline a k-decrementing service or a k-busy period service [4]. These flexible service disciplines with controllable parameters are effective for the performance optimization and applicable to routing schemes in the telecommunication systems, since these enable us to control efficiently service qualities in telecommunication systems, e.g., response time, overflow probability, etc. [5-71. There have been some analytical studies of the cyclic-service tandem queueing models, but the subject is not as well-investigated as polling models (multiqueueing models served by a single server [S]). Nair [9,10] and Taube-Netto [ll] considered a two-stage tandem queue with Poisson arrival and general service times, denoted in this paper by the Kendall notation M/G1 - Gz/l for simplicity, which is served according to an exhaustive service. Nair [12] and Katayama [2] analyzed the M/G1 - Gs/l queue with limited service (also called a nonexhaustive service and a nonzero switching rule). Katayama [13] considered an M/G1 - Gs/l queue with gated service.
131
132
T. KATAYAMAAND
K. KOBAYASHI
Nishida et al. (141, Kijnig et al. [15], and Katayama [16] studied the multistage tandem queue, with several service disciplines. There are a few studies treating the M/G1--Gz-+..-G~/l, tandem queue with multiclsss messages [17,18]. In this paper, we analyze an M/G1 - Gz/l
queue with general decrementing service of a
parameter k, where 1 < k 5 00.
The general decrementing service discipline with a parameter k = 1 corresponds to the (pure) decrementing service [19,20], also called the semi-exhaustive service [4,21], whereas if k = 00, the service discipline to the well-known exhaustive service. The results are applicable to the performance analysis of packet processing programs used in packet switching systems [11. The rest of this paper is organized as follows: Section 2 describes a two-stage tandem queue with general decrementing service and introduces some notations. Section 3 determines genSection 4 derives the Laplace-Stieltjes erating functions for joint queue-length distributions. transforms (LSTs) of message sojourn time distributions and some inequalities for the mean waiting times. Section 5 gives concluding remarks.
2. CYCLIC-SERVICE
TANDEM
QUEUEING
MODEL
This section presents a single-server two-stage tandem queueing model with general decrementing service of parameter k, where 1 5 k 5 co, and introduces some notations. The queueing system is composed of two service stages connected in series. The first stage has a queue, Qi, with a service counter, Si, and the second stage has a queue, Q2, with a service counter, S2. Messages arrive at Qi according to a Poisson process with rate A. Each message requires exactly two services before leaving the queueing system. That is, after completion of the service in Si, the message goes to Q2 to receive the service in 55, and after service-completion in 5’2, the message leaves the system. Messages in each queue are served in the order of their arrivals (FIFO). Service time 7, at each counter S,,, n = 1,2, is a random variable with general distribution function H*(t), with finite first and second moments h, and h?), respectively. The LST of H,(t) is denoted by H:(s), n = 1,2. Messages in Qi and Q2 are served by a single server in accordance with the following service discipline: after switching over to Si, the server continues serving messages in Qi until either it becomes empty, or the number of messages decreases to k less than that found upon the server’s last arrival at Sr, whichever occurs first, i.e., the k-decrementing service. Just after service completion at Sr, the server switches over to Sz and messages in Qs are served until Qs becomes empty. Two queues are served by a single server who moves among the counters alternately, that is, Sr -+ S2 --t Sr + S2 + . - +. The server’s walking time needed to switch service from one counter to another is assumed to be zero. When no messages are present in the system, the server waits for a new arrival at Sr. For simplicity, the following notation is introduced:
p,, := Ah,,
n = 1,2,
,c:=~r+~z,
h:=hl+hz.
(1)
Server utilization p = Ah is assumed to be less than unity (p < 1) to ensure stability. We denote by qn(m) the probability that m messages arrive at Qr during a service time rn, n = 1,2 and by Qn(z)( := ~~~oq~(m)~m) the generating function for qn(m). Then, we have &n(z) = fC{W
- x:>),
n = 1,2.
(2)
We also denote by g(j; m) the probability that j messages are served at Sr during an m-busy period, which corresponds to a busy period started with m messages at Sr, and by G(x; m)( := C$ g(j; m)zj) the generating function for g(j; m). From the busy-period analysis in the standard M/G/l queue [22], we then have G(z; m) = {G(z)}~,
G(z) := G(z; l),
G(Z) = zH,*{X(l - G(X))}.
m=1,2,3
,...,
(3a) (3b)
Sojourn Time Analysis
3. QUEUEING
133
ANALYSIS
This section considers two kinds of queue-length distributions in Q1 and Qs for the two-stage tandem queue with general decrementing service. It is assumed in the following sections that k is finite; i.e., 1 5 k < co and empty sums are zero. 3.1. Queue-Length
Generating Function at Service-Completion
Points
We first need to determine a set of generating functions for the joint queue-length distribution at server arrival points at Si and S2. We introduce: the steady-state probability that the server has arrived at S,,, n = 1,2, and i and
&(i,j):
j messages are waiting in Qi and Q2, respectively, i, j = 0, 1,2,. . . , and
(4 In order to derive balance equations for the queue-length distribution {&.,(i,j)}, duce a probability and its generating function:
we also intro-
f(m; j): the probability that m messages arrive at &i during the service period during which jmessagesareservedatS2,m=0,1,2
where HP’(t)
,...,
j=l,2,3
,...,
is the j-fold convolution of Hz(t) with itself,
and
F(z; j) := g
f(m;j)P, m=O
IZI I 1.
(gb)
Then, considering the events that occur during two successive server arrival points at Si and SZ, we get the following functional relationship by using (38) and the relationship F(z; j) = {Qz(z)}j:
k-l
k-l a2(z,
y) = &(O,O)G(y)
+ c
4i(i, O)G(Y)~+
i=l
@1(x, 0) - c
6di,‘$ri
i=o
y. I
(6b)
It is necessary to determine the unknown function G(y) and the unknown probabilities +i(i,O), i=O,l,..., 3.1.1.
k - 1, on the right-hand side of (6b).
Determination
of G(z)
From (3b) and Tak&cs’ lemma [22], an explicit expression for G(z) is given by
(7)
where HP’(t)
denotes the jth iterated convolution of Hi(t) with itself.
134
3.1.2.
T. KATAYAMAAND K. KOBAYASHI
of $1 (i, 0), i = 1,2, . . . , k - 1
Determination
Eliminating @~{z,Qz(z))
from (6a) and (6b) aft er setting y = Qz(z) in (6b), we get 1 4164 0) (W~d4) - G{Q&)l”} @‘(‘,O) = Zk _ G{Qz(x)}k [
+:4&o)
{zk~{Q2W
(8)
- S’G{Q+)}‘}].
i=l
is shown by Tak&cs’ lemma [22] that the denominator on the right-hand side of (8), zk G{Q2(~)}k, has exactly (k - 1) zero points z = w,., T = 1,2,. . . , k - 1, in the unit circle 1x1< 1 under the condition p 5 1 (see the Appendix). From the regularity of $(t, 0), the numerators on the right-hand side of (8) should be equal to zero for z = w,, T = 1,2,. . . , k - 1. We thus get the following linear equations for &(i,O), i = 1,. . . , k - 1: It
k-l
r=l,2
~cui(~~)~l(i,O)=(~o(w,)~~(O,O),
,...,
k-l,
i=l
where
ao (w,) := 1 - G(Q2 (w,)), Eli
(ur) := 4 - G i&2 (4)*,
i=1,2
,...,
(9b)
k-l.
Using Cramer’s formula, &(i, 0)) i = 1,2,. . . , k - 1 can be given by an expression with unknown probability & (0,O) as follows: 41(&O) = &(O,O)Ai,
I Ai := -1 ‘PDi
i=1,2
,...,
k-l,
(10)
where IDil and 101 are the determinants formed by coefficients of the simultaneous equations (9a). The unknown probability &(O, 0) in (10) shall be determined in the next section. 3.1.3.
Determination
of &(O, 0)
Substituting &(i, 0) = &(O, O)Ai, i = 1,2,. . . , k - 1, for +1(&O) on the right-hand side of (8)) $(s, 0) can be expressed by a form with only unknown probability (PI(0,O). As a result, @pz(z,y) given by (6b) can also be expressed by a form with only unknown probability &(O, 0). The probability & (0,O) should be determined by the normalization condition, @I(1,O) + @2(1,l) = 1. We thus get
l+$-++E()I 1-l
--I
Ai
(11)
.
r=l
In this way, the generating functions &(z, y), n = 1,2, have been completely determined. thus obtain the following results.
We
1. The generating functions @n(z) y), n * 1,2, for the joint queue-length distribution {4n(i, j)} are given by:
THEOREM
@l(Z)
0) =
4lP, zk -
0)
G{Q~(z>)~
~kG{Q2(41
- G{Q2Wk
k-l + c
Ai
{~“G{Qz(~}~
- ~“G{Q~(z))~}
i=l Q2b,Y)
=
41(0,0) xk -
G{Q&)}”
G(Y) {xk -
G{QzWk}
(124
1 ,
- G(Y)” + G{Q&))G(Y)~
k-l -I- c i=l where
L& { G(y)“G{Q2(x)}’
-
q#cq&(~)}~
+ x”G(y)$
-
xiG(y)k}
, (12b) I
G(x), Ai, i = 1,2, . . . , k - 1, and &(O, 0) are given by (7)) (10)) and (II), respectively.
1
Sojourn Time Analysis
Queue-Length
3.2.
Generating
Function
135
at Message
Departure
Points
We next analyze queue-length distributions at departure points of messages from each service counter. We introduce: nl(i):
the steady-state probability that i messages are waiting in &I just after a message has completed service at the first stage, i = 0, 1,2,. . . ,
7rg(i,j):
the steady-state probability that i messages are waiting in &I and j messages are waiting in Q2 just after a message has completed service at the second stage, i, j = 0, 112, . . . ,
and l-II(Z) := 5 7rl(i)Zi, i=o l&(2,9)
:= 2
2
id) j&l
14 51,
(13)
?r2(i, j)z”yj,
I4
Id 5 1.
(14
In order to derive a balance equation for the queue-length distribution (~1 (i)}, we also introduce a conditional probability and its generating function: q(i; m): the probability that during an m-busy period at &, i messages are waiting in &I just after a message has completed its service, i = 0, 1,2,. . . , m = 1,2,3,. . . , and Q(z; m) := 2 q(i; m)xi, i=o
1x1 I 1.
(15)
Then, considering the events that occur during two successive message departure points at the service counters S1 and S2, we have the following equations: k-l al(X)
=
Cl
o)Qb; 1) + c
$I@,
&Cm, O)Q(x; m>
m=l
9
+
HIz(x,y) =
i
91(x, 0) -
‘2
C2Q2(x) @2(x, y) + Qao y Y
,
&(m,O)xm
m=O
(164
II
Pz(x:,
y) - JJ2(x, 0)))
(16b)
where Cl and C2 are normalizing constants. After a lengthy but usual analysis, we obtain the following equations of the generating functions Q(z; m) on the right-hand side of (Isa),
&km)
=
Hi(x(1 - X)1 Hf{X(l - X)}
(xm _ 1),
m=l,2,3
X -
The constants Cl and CZ can be determined by the normalization IIz(l,l) = 1 and the relationship IIz(x,O)/II2(1,0) = 91(x,O)/@l(l,O). c-l 1
c-1 2
=
1 -p1 _
k
~le40)
dlW)* 1-P
l-
k +
Wl(O,O)
,....
(17)
conditions, II1 (1) = We thus have
1,
k-l -C&-m)& m=l
,
(1%
I Pb)
In thii way, the generating functions l&(x) and II2 (x, y) have been completely determined. We thus obtain the following results.
136
T. KATAYAMA AND K. KOBAYASHI
THEOREM
2. The
{nr(i)} and {4&j)},
generating functions III(Z) and l&(z, y) for the queue-length respectively, are given by:
distributions
(lgb) I
4. SOJOURN
TIME ANALYSIS
Let &.,, n = 1,2, denote the sojourn time in the nth stage (i.e., the time spent by a message in the nth stage) and denote by 63,(t) the distribution function of 0,. Similarly, denote by w, and W,,(t), n = 1,2, the waiting time in the nth queue and the distribution function of w,, respectively. In addition, let &3(t) denote the total sojourn time (the time spent by a message in the queueing system) and let 0(t) be its distribution function. Since a message departing from Si will leave behind m (= 0, 1,2,. . . ) messages which arrived during its sojourn time 01 because of the FIFO queueing discipline, the following equation for &(t) is obtained:
c Jmy-e-%el(t)= Cm
l&(x).
Xrn
0
m=O
(204
.
A similar equation holds for Cl(t),
c J00ye+ 00
Xrn
m.
0
m=O
de(t) = &(x,x).
G’Ob)
Thus, we obtain the following result. 3. The LST e;(s) time 8 are given by:
THEOREM
e;(s)
=
for the sojourn time 8r and the LST e*(s) for the total sojourn
Clh(O,OPi(s) S -
X(1 - H,*(S)) [1+:-(&J*
+~l~m{~-(~)k-m}-{~-(~)k}@l&S/o;lo)], (214 e*(S)
=
C2wxs) s-
X(1 - H,*(s))
@lb) I
{~1(1-~,0)-Q2(l-~,1-~)}.
We derive explicit expressions for mean waiting times. The mean sojourn times, E(Bi) and E(8), can be obtained by letting s + 0 after differentiating e;(s) and e*(s) with respect to s, respectively. Using the relationships E(Wn)k = E(@,& - h,, n = 1,2, and E(e)k = E(6$)1, + E(6&, we get the following results. COROLLARY
1. The mean message waiting times, E(W&, E(wl)k
= -
=
-
(224
hl,
a=0 E(W2)k
-
n = 1,2, are obtained by:
[g
{e*(S)
-
-
e;(S)}] e=O
h2.
(22b)
Sojourn Time Analysis For a special case of k = 1 (the semi-exhaustive
E(W&i
= - %I+ 1-
2(1
Pl
-
P
P112
137
service [4]), E(Wn)k=l,
n = 1,2, are given by:
(234 (23b)
Ah?).
The mean total sojourn time E(@k=i is given by:
E(e)k=l = i -
’
Pl
1 -P1(1+
PlP2
h+ (1 -
P2)
Xh$2)
P) h2 + 2 (1 - pl)2 (1 - p)
Pl) (1 -
+
1
--Xh$? 2(1 -P)
(23~) I
We next consider upper and lower bounds of mean waiting times E(Wn)k, n = 1,2. First, we recall the conservation law for the cyclic-service tandem queueing model [4], PE
(wl)k
+ p2ECW2)k
=
w
XP
{
hy) + hp) + 2hlhz)
.
(24
-P>
Comparing two service disciplines at the service counter Si, i.e., the k-decrementing the (k + 1)-decrementing service, we have
service and
(25) since the mean waiting time in the second stage should be monotonically increasing as the parameter k increases, while, as also shown from (24), the mean waiting time in the first stage should be decreasing consequently [16]. Using the conservation law (24), the inequality (25) and the results for k = CO (the exhaustive service [ll,lS]), we thus obtain the following upper and lower bounds of the mean waiting times. COROLLARY2. (26) where x (1 - Pl) E(W1)k=cO= E (Wdk=m
(l-pl~~)(I-p)h~+2(1-Pl+p2)(l_P) = +-$I
+
2 (1
-
(274
PIP (1
-
Pl + P2)
XP +
{ hy) + hg) } ,
Pl + P2)
(I{
(1
-
P)
p) h2
hr) + hf) } .
(27b)I
REMARK 4.1. It is easily confirmed that the mean waiting times E(Wn)k=i and E(Wn)kzm, n = 1,2, satisfy the conservation law given by (24). It is also shown from (23a) and (27a) that the inequality, derived as a special case of (25), E(Wr)k=r > E(wr)k=, always holds under the stability condition p < 1. I
5. CONCLUDING
REMARKS
The study of cyclic-service tandem queueing models is important to design various call processing programs used in communication network nodes. We have presented a flexible service discipline with general decrementing service. From theoretical points of view, we have analyzed the basic two-stage tandem queueing model. Using the solution (i.e., zero points in a unit circle) of a well-known transcendental equation, we have determined the generating functions for joint
T. KATAYAMA AND K. KOBAYASHI
138
queue-length distributions and the LSTs for sojourn time distributions. As a result, we have also derived explicit expressions, simple inequalities and upper and lower bounds for mean waiting times in each stage, though Further
called walking of switching obtained
time or overhead
proof is given for the propriety
to the evaluation
time)
of the influence
on some performance
rules [6,7,18], and decrementing-service
here, it may be possible
a cost function computer
no rigorous
research will be extended
defined
to determine
by a linear
combination
measures
parameters.
an optimal
of (25).
of the switch-over
For example,
using the results
k = kept so as to minimize
parameter
of mean waiting
time (also
and to the optimization
times
in each stage by help of
programs,
APPENDIX ZEROS
OF THE DENOMINATOR
OF EQUATION
(8)
From (2), we have zk - G {Qs(z)}~ Let us consider
a transcendental
equation
2k - H*{X(l
-z)}
= xk - G [H; (X(1 -
x)}]” .
(AlI
for 2, = 0,
H*(s)
:== G{H;(s)}“.
(AS)
Applying TaJ&s’ lemma [22] to (A2), we obtain the following results: if p I 1, then equation (A2) has exactly k roots wr, T = 1,2,. . . , k - 1, in the unit circle 1x1 < 1, while Wk = 1 and wt-, k - 1 are explicitly expressed by r=l,2,..., Co (-X)j-10;
W T
:=
c
j=l
j!
(-g--l dXj-1
{HeWk
(A3)
1
where i=J-1,
r=1,2
,,..,
k-l.
(A4
REFERENCES 1. F. Ishino et al., High-throughput packet switching system, Review of the Electrical Communications Laboratories, NTT 54 (6), 689696 (1986). 2. T. Katayama, Analysis of a finite intermediate waiting room tandem queue attended by a moving server with walking time, tins. of IECE of Japan E-64 (9) 571-578 (1981). 3. H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation Vol. 1, Vacation and Priority S@ems, Part 1, Elsevier Science Publisher B.V., North-Holland, (1991). 4. T. Katayama, A cyclic-service tandem queueing model with semi-exhaustive service, In Performance of Distributed and Parallel Systems, (Edited by T. Hssegawa, H. Takagi and Y. Takahashi), pp. 173-183, North-Holland, (1989). 5. S.S. Fratini, Analysis of dynamic priority queue, Commun. Statist.-Stochastic Models 6 (3), 415-444 (1990). 6. M. Hofri and K.W. Boss, On the optimal control of two queues with server set-up times and its analysis, SIAM J. Comput. 16 (2), 399420 (1987). 7. T.L. Lai and Z. Ymg, Open bandit processes and optimal scheduiing of queueing networks, Adv. A&. Prob. 20 (2), 447-472 (1988). a. H. bevy and M. Sidi, Polling systems: Applications, modeling, and optimisation, IEEE tins. on Commun. 88 (lo), 1750-1760 (1990). 9. S.S. Nair, Semi-Markov analysis of two queues in series attended by a single server, Bull. Sot. Math. Belgique 22, 355-367 (1970). 10. S.S. Nair, Two queues in series attended by a single server, Bull. Sot. Math. Belgique 25, 160-176 (1973). 11. M. ‘&ube-Netto, Two queues in tandem attended by a single server, Oper. Res. 25 (l), 140-147 (1977). 12. S.S. Nair, A single server tandem queue, J. Appl. Prob. 8 (1) 95-109 (1971). 13. T. Katayama, Analysis of a tandem queueing system with gate attended by a moving server, Review of the Electrical Communications Laboratories, NTT 29 (3/4), 254-267 (1981). 14. T. Nishida, Y. Serikawa and K. Yoneyama, n queues in tandem attended by a single server, Rep. Stat. Appl. Res. JUSE 26 (l), 14-19 (1979).
Sojourn Time Analysis
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15. D. K&rig and V. Schmit, Relationships between time/customer stationary characteristics of tandem queues attended by a single server, J. Oper. Res. Sot. Japan 27 (3), 191-204 (1984). 16. T. Katayama, Mean sojourn times in a multi-stage tandem queue served by a single server, J. Oper. Rea. Sot. Japan 31 (2), 233-250 (1988). 17. T. Katayama, A cyclic-service tandem queueing model with parallel queues in the first stage, Commun. Stat&.-Stochastic Models 4 (3), 421-443 (1988). 18. T. Katayama, Performance analysis and optimization of a cyclic-service tandem queueing system with multi-class customers, Computers Math. Applic. 24 (l/2), 25-34 (1992). Cybernetics 19. B.G. Pittel’, Optimal control in a mass service system with several flows of demands, Engineering 11 (6), 1029-1044 (1972). ‘84, Models of 20. H. Takagi, Mean message waiting time in a symmetric polling system, In Performance Computer System Performance, (Edited by Gelenbe), pp. 293-302, North-Holland, (1985). 21. J.W. Cohen, A two-queue model with semi-exhaustive alternating service, In Performance ‘87, (Edited by P.-J. Courtois and G. Latouche), pp. 19-37, North-Holland, (1988). 22. L. Takks, Introduction to the Theory of Queues, Oxford University Press, New York, (1962).
Pergamon
Modelling
Vol. 22, No. 19-12,
pp. 131-139,
1995
Copyright@1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 08957177/95 $9.50 + 0.99
0895-7177(95)00189-l
Sojourn Time Analysis for a Cyclic-Service Tandem Queueing Model with General Decrementing Service T.
KATAYAMA AND K. KOBAYASHI Faculty of Engineering Toyama Prefectural University Kosugi-Machi, Toyama 939-03, Japan
Abstract-A sojourn time analysis is provided for a cyclic-service tandem queue with general decrementing service which operates as follows: starting once a service of queue 1 in the first stage, a single server continues serving messages in queue 1 until either queue 1 becomes empty, or the number of messages decreases to k lees than that found upon the server’s last arrival at queue 1, whichever occurs first, where 1 5 k < 00. After service completion in queue 1, the server switches over to queue 2 in the second stage and serves all messages in queue 2 until it becomes empty. It is assumed that an arrival stream is Poissonian, message service times at eech stage are generally distributed, and switch-over times are zero. This paper analyzes joint queue-length distributions and message sojourn time distributions. Keywords-General tion.
decrementingservice,Cyclic-servicetandemqueue, Sojourn time distribu-
1. INTRODUCTION There are several practical examples of tandem queueing models served by a single server in the performance evaluation of computer operating systems and call processing in switching systems in telecommunication networks [1,2]. This paper considers such a tandem queueing model with general decrementing service of a parameter k which operates as follows [3]. For the moment, suppose that messages in the first stage are served. A single server continues serving messages in queue 1 until either queue 1 becomes empty, or the number of messages decreases to k less than that found upon the server’s last arrival at queue 1, whichever occurs first. We also call this service discipline a k-decrementing service or a k-busy period service [4]. These flexible service disciplines with controllable parameters are effective for the performance optimization and applicable to routing schemes in the telecommunication systems, since these enable us to control efficiently service qualities in telecommunication systems, e.g., response time, overflow probability, etc. [5-71. There have been some analytical studies of the cyclic-service tandem queueing models, but the subject is not as well-investigated as polling models (multiqueueing models served by a single server [S]). Nair [9,10] and Taube-Netto [ll] considered a two-stage tandem queue with Poisson arrival and general service times, denoted in this paper by the Kendall notation M/G1 - Gz/l for simplicity, which is served according to an exhaustive service. Nair [12] and Katayama [2] analyzed the M/G1 - Gs/l queue with limited service (also called a nonexhaustive service and a nonzero switching rule). Katayama [13] considered an M/G1 - Gs/l queue with gated service.
131
132
T. KATAYAMAAND
K. KOBAYASHI
Nishida et al. (141, Kijnig et al. [15], and Katayama [16] studied the multistage tandem queue, with several service disciplines. There are a few studies treating the M/G1--Gz-+..-G~/l, tandem queue with multiclsss messages [17,18]. In this paper, we analyze an M/G1 - Gz/l
queue with general decrementing service of a
parameter k, where 1 < k 5 00.
The general decrementing service discipline with a parameter k = 1 corresponds to the (pure) decrementing service [19,20], also called the semi-exhaustive service [4,21], whereas if k = 00, the service discipline to the well-known exhaustive service. The results are applicable to the performance analysis of packet processing programs used in packet switching systems [11. The rest of this paper is organized as follows: Section 2 describes a two-stage tandem queue with general decrementing service and introduces some notations. Section 3 determines genSection 4 derives the Laplace-Stieltjes erating functions for joint queue-length distributions. transforms (LSTs) of message sojourn time distributions and some inequalities for the mean waiting times. Section 5 gives concluding remarks.
2. CYCLIC-SERVICE
TANDEM
QUEUEING
MODEL
This section presents a single-server two-stage tandem queueing model with general decrementing service of parameter k, where 1 5 k 5 co, and introduces some notations. The queueing system is composed of two service stages connected in series. The first stage has a queue, Qi, with a service counter, Si, and the second stage has a queue, Q2, with a service counter, S2. Messages arrive at Qi according to a Poisson process with rate A. Each message requires exactly two services before leaving the queueing system. That is, after completion of the service in Si, the message goes to Q2 to receive the service in 55, and after service-completion in 5’2, the message leaves the system. Messages in each queue are served in the order of their arrivals (FIFO). Service time 7, at each counter S,,, n = 1,2, is a random variable with general distribution function H*(t), with finite first and second moments h, and h?), respectively. The LST of H,(t) is denoted by H:(s), n = 1,2. Messages in Qi and Q2 are served by a single server in accordance with the following service discipline: after switching over to Si, the server continues serving messages in Qi until either it becomes empty, or the number of messages decreases to k less than that found upon the server’s last arrival at Sr, whichever occurs first, i.e., the k-decrementing service. Just after service completion at Sr, the server switches over to Sz and messages in Qs are served until Qs becomes empty. Two queues are served by a single server who moves among the counters alternately, that is, Sr -+ S2 --t Sr + S2 + . - +. The server’s walking time needed to switch service from one counter to another is assumed to be zero. When no messages are present in the system, the server waits for a new arrival at Sr. For simplicity, the following notation is introduced:
p,, := Ah,,
n = 1,2,
,c:=~r+~z,
h:=hl+hz.
(1)
Server utilization p = Ah is assumed to be less than unity (p < 1) to ensure stability. We denote by qn(m) the probability that m messages arrive at Qr during a service time rn, n = 1,2 and by Qn(z)( := ~~~oq~(m)~m) the generating function for qn(m). Then, we have &n(z) = fC{W
- x:>),
n = 1,2.
(2)
We also denote by g(j; m) the probability that j messages are served at Sr during an m-busy period, which corresponds to a busy period started with m messages at Sr, and by G(x; m)( := C$ g(j; m)zj) the generating function for g(j; m). From the busy-period analysis in the standard M/G/l queue [22], we then have G(z; m) = {G(z)}~,
G(z) := G(z; l),
G(Z) = zH,*{X(l - G(X))}.
m=1,2,3
,...,
(3a) (3b)
Sojourn Time Analysis
3. QUEUEING
133
ANALYSIS
This section considers two kinds of queue-length distributions in Q1 and Qs for the two-stage tandem queue with general decrementing service. It is assumed in the following sections that k is finite; i.e., 1 5 k < co and empty sums are zero. 3.1. Queue-Length
Generating Function at Service-Completion
Points
We first need to determine a set of generating functions for the joint queue-length distribution at server arrival points at Si and S2. We introduce: the steady-state probability that the server has arrived at S,,, n = 1,2, and i and
&(i,j):
j messages are waiting in Qi and Q2, respectively, i, j = 0, 1,2,. . . , and
(4 In order to derive balance equations for the queue-length distribution {&.,(i,j)}, duce a probability and its generating function:
we also intro-
f(m; j): the probability that m messages arrive at &i during the service period during which jmessagesareservedatS2,m=0,1,2
where HP’(t)
,...,
j=l,2,3
,...,
is the j-fold convolution of Hz(t) with itself,
and
F(z; j) := g
f(m;j)P, m=O
IZI I 1.
(gb)
Then, considering the events that occur during two successive server arrival points at Si and SZ, we get the following functional relationship by using (38) and the relationship F(z; j) = {Qz(z)}j:
k-l
k-l a2(z,
y) = &(O,O)G(y)
+ c
4i(i, O)G(Y)~+
i=l
@1(x, 0) - c
6di,‘$ri
i=o
y. I
(6b)
It is necessary to determine the unknown function G(y) and the unknown probabilities +i(i,O), i=O,l,..., 3.1.1.
k - 1, on the right-hand side of (6b).
Determination
of G(z)
From (3b) and Tak&cs’ lemma [22], an explicit expression for G(z) is given by
(7)
where HP’(t)
denotes the jth iterated convolution of Hi(t) with itself.
134
3.1.2.
T. KATAYAMAAND K. KOBAYASHI
of $1 (i, 0), i = 1,2, . . . , k - 1
Determination
Eliminating @~{z,Qz(z))
from (6a) and (6b) aft er setting y = Qz(z) in (6b), we get 1 4164 0) (W~d4) - G{Q&)l”} @‘(‘,O) = Zk _ G{Qz(x)}k [
+:4&o)
{zk~{Q2W
(8)
- S’G{Q+)}‘}].
i=l
is shown by Tak&cs’ lemma [22] that the denominator on the right-hand side of (8), zk G{Q2(~)}k, has exactly (k - 1) zero points z = w,., T = 1,2,. . . , k - 1, in the unit circle 1x1< 1 under the condition p 5 1 (see the Appendix). From the regularity of $(t, 0), the numerators on the right-hand side of (8) should be equal to zero for z = w,, T = 1,2,. . . , k - 1. We thus get the following linear equations for &(i,O), i = 1,. . . , k - 1: It
k-l
r=l,2
~cui(~~)~l(i,O)=(~o(w,)~~(O,O),
,...,
k-l,
i=l
where
ao (w,) := 1 - G(Q2 (w,)), Eli
(ur) := 4 - G i&2 (4)*,
i=1,2
,...,
(9b)
k-l.
Using Cramer’s formula, &(i, 0)) i = 1,2,. . . , k - 1 can be given by an expression with unknown probability & (0,O) as follows: 41(&O) = &(O,O)Ai,
I Ai := -1 ‘PDi
i=1,2
,...,
k-l,
(10)
where IDil and 101 are the determinants formed by coefficients of the simultaneous equations (9a). The unknown probability &(O, 0) in (10) shall be determined in the next section. 3.1.3.
Determination
of &(O, 0)
Substituting &(i, 0) = &(O, O)Ai, i = 1,2,. . . , k - 1, for +1(&O) on the right-hand side of (8)) $(s, 0) can be expressed by a form with only unknown probability (PI(0,O). As a result, @pz(z,y) given by (6b) can also be expressed by a form with only unknown probability &(O, 0). The probability & (0,O) should be determined by the normalization condition, @I(1,O) + @2(1,l) = 1. We thus get
l+$-++E()I 1-l
--I
Ai
(11)
.
r=l
In this way, the generating functions &(z, y), n = 1,2, have been completely determined. thus obtain the following results.
We
1. The generating functions @n(z) y), n * 1,2, for the joint queue-length distribution {4n(i, j)} are given by:
THEOREM
@l(Z)
0) =
4lP, zk -
0)
G{Q~(z>)~
~kG{Q2(41
- G{Q2Wk
k-l + c
Ai
{~“G{Qz(~}~
- ~“G{Q~(z))~}
i=l Q2b,Y)
=
41(0,0) xk -
G{Q&)}”
G(Y) {xk -
G{QzWk}
(124
1 ,
- G(Y)” + G{Q&))G(Y)~
k-l -I- c i=l where
L& { G(y)“G{Q2(x)}’
-
q#cq&(~)}~
+ x”G(y)$
-
xiG(y)k}
, (12b) I
G(x), Ai, i = 1,2, . . . , k - 1, and &(O, 0) are given by (7)) (10)) and (II), respectively.
1
Sojourn Time Analysis
Queue-Length
3.2.
Generating
Function
135
at Message
Departure
Points
We next analyze queue-length distributions at departure points of messages from each service counter. We introduce: nl(i):
the steady-state probability that i messages are waiting in &I just after a message has completed service at the first stage, i = 0, 1,2,. . . ,
7rg(i,j):
the steady-state probability that i messages are waiting in &I and j messages are waiting in Q2 just after a message has completed service at the second stage, i, j = 0, 112, . . . ,
and l-II(Z) := 5 7rl(i)Zi, i=o l&(2,9)
:= 2
2
id) j&l
14 51,
(13)
?r2(i, j)z”yj,
I4
Id 5 1.
(14
In order to derive a balance equation for the queue-length distribution (~1 (i)}, we also introduce a conditional probability and its generating function: q(i; m): the probability that during an m-busy period at &, i messages are waiting in &I just after a message has completed its service, i = 0, 1,2,. . . , m = 1,2,3,. . . , and Q(z; m) := 2 q(i; m)xi, i=o
1x1 I 1.
(15)
Then, considering the events that occur during two successive message departure points at the service counters S1 and S2, we have the following equations: k-l al(X)
=
Cl
o)Qb; 1) + c
$I@,
&Cm, O)Q(x; m>
m=l
9
+
HIz(x,y) =
i
91(x, 0) -
‘2
C2Q2(x) @2(x, y) + Qao y Y
,
&(m,O)xm
m=O
(164
II
Pz(x:,
y) - JJ2(x, 0)))
(16b)
where Cl and C2 are normalizing constants. After a lengthy but usual analysis, we obtain the following equations of the generating functions Q(z; m) on the right-hand side of (Isa),
&km)
=
Hi(x(1 - X)1 Hf{X(l - X)}
(xm _ 1),
m=l,2,3
X -
The constants Cl and CZ can be determined by the normalization IIz(l,l) = 1 and the relationship IIz(x,O)/II2(1,0) = 91(x,O)/@l(l,O). c-l 1
c-1 2
=
1 -p1 _
k
~le40)
dlW)* 1-P
l-
k +
Wl(O,O)
,....
(17)
conditions, II1 (1) = We thus have
1,
k-l -C&-m)& m=l
,
(1%
I Pb)
In thii way, the generating functions l&(x) and II2 (x, y) have been completely determined. We thus obtain the following results.
136
T. KATAYAMA AND K. KOBAYASHI
THEOREM
2. The
{nr(i)} and {4&j)},
generating functions III(Z) and l&(z, y) for the queue-length respectively, are given by:
distributions
(lgb) I
4. SOJOURN
TIME ANALYSIS
Let &.,, n = 1,2, denote the sojourn time in the nth stage (i.e., the time spent by a message in the nth stage) and denote by 63,(t) the distribution function of 0,. Similarly, denote by w, and W,,(t), n = 1,2, the waiting time in the nth queue and the distribution function of w,, respectively. In addition, let &3(t) denote the total sojourn time (the time spent by a message in the queueing system) and let 0(t) be its distribution function. Since a message departing from Si will leave behind m (= 0, 1,2,. . . ) messages which arrived during its sojourn time 01 because of the FIFO queueing discipline, the following equation for &(t) is obtained:
c Jmy-e-%el(t)= Cm
l&(x).
Xrn
0
m=O
(204
.
A similar equation holds for Cl(t),
c J00ye+ 00
Xrn
m.
0
m=O
de(t) = &(x,x).
G’Ob)
Thus, we obtain the following result. 3. The LST e;(s) time 8 are given by:
THEOREM
e;(s)
=
for the sojourn time 8r and the LST e*(s) for the total sojourn
Clh(O,OPi(s) S -
X(1 - H,*(S)) [1+:-(&J*
+~l~m{~-(~)k-m}-{~-(~)k}@l&S/o;lo)], (214 e*(S)
=
C2wxs) s-
X(1 - H,*(s))
@lb) I
{~1(1-~,0)-Q2(l-~,1-~)}.
We derive explicit expressions for mean waiting times. The mean sojourn times, E(Bi) and E(8), can be obtained by letting s + 0 after differentiating e;(s) and e*(s) with respect to s, respectively. Using the relationships E(Wn)k = E(@,& - h,, n = 1,2, and E(e)k = E(6$)1, + E(6&, we get the following results. COROLLARY
1. The mean message waiting times, E(W&, E(wl)k
= -
=
-
(224
hl,
a=0 E(W2)k
-
n = 1,2, are obtained by:
[g
{e*(S)
-
-
e;(S)}] e=O
h2.
(22b)
Sojourn Time Analysis For a special case of k = 1 (the semi-exhaustive
E(W&i
= - %I+ 1-
2(1
Pl
-
P
P112
137
service [4]), E(Wn)k=l,
n = 1,2, are given by:
(234 (23b)
Ah?).
The mean total sojourn time E(@k=i is given by:
E(e)k=l = i -
’
Pl
1 -P1(1+
PlP2
h+ (1 -
P2)
Xh$2)
P) h2 + 2 (1 - pl)2 (1 - p)
Pl) (1 -
+
1
--Xh$? 2(1 -P)
(23~) I
We next consider upper and lower bounds of mean waiting times E(Wn)k, n = 1,2. First, we recall the conservation law for the cyclic-service tandem queueing model [4], PE
(wl)k
+ p2ECW2)k
=
w
XP
{
hy) + hp) + 2hlhz)
.
(24
-P>
Comparing two service disciplines at the service counter Si, i.e., the k-decrementing the (k + 1)-decrementing service, we have
service and
(25) since the mean waiting time in the second stage should be monotonically increasing as the parameter k increases, while, as also shown from (24), the mean waiting time in the first stage should be decreasing consequently [16]. Using the conservation law (24), the inequality (25) and the results for k = CO (the exhaustive service [ll,lS]), we thus obtain the following upper and lower bounds of the mean waiting times. COROLLARY2. (26) where x (1 - Pl) E(W1)k=cO= E (Wdk=m
(l-pl~~)(I-p)h~+2(1-Pl+p2)(l_P) = +-$I
+
2 (1
-
(274
PIP (1
-
Pl + P2)
XP +
{ hy) + hg) } ,
Pl + P2)
(I{
(1
-
P)
p) h2
hr) + hf) } .
(27b)I
REMARK 4.1. It is easily confirmed that the mean waiting times E(Wn)k=i and E(Wn)kzm, n = 1,2, satisfy the conservation law given by (24). It is also shown from (23a) and (27a) that the inequality, derived as a special case of (25), E(Wr)k=r > E(wr)k=, always holds under the stability condition p < 1. I
5. CONCLUDING
REMARKS
The study of cyclic-service tandem queueing models is important to design various call processing programs used in communication network nodes. We have presented a flexible service discipline with general decrementing service. From theoretical points of view, we have analyzed the basic two-stage tandem queueing model. Using the solution (i.e., zero points in a unit circle) of a well-known transcendental equation, we have determined the generating functions for joint
T. KATAYAMA AND K. KOBAYASHI
138
queue-length distributions and the LSTs for sojourn time distributions. As a result, we have also derived explicit expressions, simple inequalities and upper and lower bounds for mean waiting times in each stage, though Further
called walking of switching obtained
time or overhead
proof is given for the propriety
to the evaluation
time)
of the influence
on some performance
rules [6,7,18], and decrementing-service
here, it may be possible
a cost function computer
no rigorous
research will be extended
defined
to determine
by a linear
combination
measures
parameters.
an optimal
of (25).
of the switch-over
For example,
using the results
k = kept so as to minimize
parameter
of mean waiting
time (also
and to the optimization
times
in each stage by help of
programs,
APPENDIX ZEROS
OF THE DENOMINATOR
OF EQUATION
(8)
From (2), we have zk - G {Qs(z)}~ Let us consider
a transcendental
equation
2k - H*{X(l
-z)}
= xk - G [H; (X(1 -
x)}]” .
(AlI
for 2, = 0,
H*(s)
:== G{H;(s)}“.
(AS)
Applying TaJ&s’ lemma [22] to (A2), we obtain the following results: if p I 1, then equation (A2) has exactly k roots wr, T = 1,2,. . . , k - 1, in the unit circle 1x1 < 1, while Wk = 1 and wt-, k - 1 are explicitly expressed by r=l,2,..., Co (-X)j-10;
W T
:=
c
j=l
j!
(-g--l dXj-1
{HeWk
(A3)
1
where i=J-1,
r=1,2
,,..,
k-l.
(A4
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