In [5,6,14], - J-Stage

The Operations Research Society of Japan

NII-Electronic Library Service

The OpeiationsReseaich Society of Japan

Journal of the Operations Research Society of JapanVol, 56, No. 2, June 2013, pp, 111-136

@ The Operatiens Research Society of Japan

THE MULTI-CLASS FIFO M/G/1 QUEUEWITH EXPONENTIAL WORKING VACATIONS

Ybshiaki lnoue Tletsuya Takine

0saka Uitiversity

CReceived March 10, 2012; Revised January 1, 2013))

Abstract We consider a statiDnary multi-class FIFO M/Gfl queue with exponential,working vacations,

where a server works at two different processing rates. There are K classes of customers, and the arrival

rates and the distributions of the amount of service requirements of arriving customers depend on boththeir customer classes and the server state. When the system becomes empty, the server takes a working

vacation, during which customers are served at processing rate 7 (or > O), If the system is empty at theend of the working vacation, the server takes another working vacatioll. On the other hand, if a customer is

being served at the end of the working vacation, the server switches its processing rate to one and continues

to serve customers in a preemptive-resume manner, until the system becomes empty. FbT this queue, we

derive various quantities of interest, including the Laplace-Stieltjes transfbrms of the actual waiting time

and sojourn time distributions, and the joint transform of the numbers of customers and the amounts

of unfinished work in respective crasses, As by-products, we also obtain various results of a stationary

multi-class FIFO MIGII queue with Poisson disasters.

Keywords: Queue, M/G/1, multi-class, FIFO, working vacations, disasters

1. Introduction

This paper considers a single-server queue with working vacations. In queues with working

vacations, the server takes a working vacation when the system becomes empty. Contraryto ordinary vacations models, customers are served at processing rate 7 (or > O), which may

differ from the normal processing rate of one. If a customer is being served at the end of

the working vacation, the server switches its processing rate to'one and continues to serve

customers, until the system becomes empty.

The queueing model with working vacations was first introduced in [9], as a model

of an access router in a reconfigurable wavelength division multiplexing (WDM) optical

access netwoTk. While each access router has its own wavelength, there are some additional

wavelengths that are shared among several access routers, and those additional wavelengths

are assigned to those access routers cyclically. A working va £ ation period then corresponds

to the situation that the access router has no additional wavelengths and the fb]lowing

period with the normal processing rate of one corresponds to the situation that Phe accessrouter utilizes the additional wavelengths as well. In [9], an M/M!1 queue with exponential

working vacations is studied. In [5,6,14], the medel of [9] is generalized to the M!Gll

queue.

In current communication networks, input traffic is usually a superposition of several

packet streams such as video, audio, and data traMc, which have different arrival rates and

packet length distributions, We thus consider a model with several classes of customers so

that such a feature can be incorporated.

lll




112 Z Inoue & T. 7Zikine

Note that queues with working vacations are also applicable to modeling a class of trafic

engineering schemes. IJbr example, we consider the fo11owing scenario. The network system

provides a fixed, primary route for each destination. When packet transmissions start on

the primary path, the network system tries to find the lightly-loaded second path, and ifsuch a path is found after some delay (and the sender node still transmits packets), someof packet streams served on the primary path wil! be re-routed to the second path, In thisscenario, the working vacation period corresponds to the interval during whieh the system is

seeking the second path, and the fo11owing normal service period corresponds to the intervalafter re-routing.

rlb

model this scenario, we set 7 to be one, while the arrival rate in thenormal service period is less than that in the working vacation. We thus generalize theconventional model with werking vacations and assume that the arrival rate in the working

vacation and normal service periods may be different.

Past studies on the M/G/1 queues with working vacations take an approach that the

queue length process is analyzed first and then other performance measures of the model

are derived from the result of the queue length, Furthermore, to make the analysis of the

queue length simple, those studies assume the preemptive-repeat with resampling when

working vacations end, i.e,, the server always restarts the ongoing service at the beginningof normal service periods, where the new service time is resampled according to the service

time distribution. On the other hand, in our model, the server continues the ongoing service

at the beginning of a normal service period in a preemptive-resume manner.

In general, the queue length process in multi-class FIFO queues is not easy to analyzedirectly [1, 10]. Therefore, we first analyze the stationary amount of work in system and

obtain its LST. Using this result, we derive thejoint LST of the attained waiting time [8] and

the remaining service requirement in terms of the LST of work in system. Because t･he serverhas two different processing rates, the analysis of the attained waiting time distribution inour model is not as simple as in [1,10]. This also makes the joint LST of the attained

waiting time and the remaining service requirement complicated. We classify the attained

waiting time into several cases, so that the fbrmula fbr the joint LST of the attained waiting

time and the remaining service requirement is given in a comprehensible form.

Note that all waiting customers in the FIFO system arrived during the attained waiting

time [1, 10]. Based on this observation, we obtain the joint transfbrm of the queue lengthsand the amounts of work in system in respective classes, which is the main result of this

paper. We also derive the LSTs of the stationary distributions of waiting time and sojourn

time and the joint transform of the length of a randomly chosen busy cycle and the numberQf customers served in the cycle.

Owing to the independent and stationary increment of Poisson arrival processes, thestationary system behavior conditioned that the server is on working vacation is equivalentto that in the corresponding queue with disasters. Therefbre, as by-products, we also obtain

various fbrmulas for the multi-class FIFO M/G/1 queue with Poisson disasters.

The rest of this paper is organized as fo11ows. In section 2, we describe the mathematicalmodel. In section 3, the stationary amount of work in system is analyzed, In section 4,the actual waiting time and sojourn time distributions are analyzed. In section 5, we study

the joint distribution of the numbers of customers and the arnounts of work in system inrespective clasSes. In section 6, we analyze the busy cycle, Finally, some concluding remarks

are provided in section 7.




FIFO MIGII eueue with Vilorking Y,2ications l13

2. Model

We consider a stationary multi-class FIFO M/G/1 queue with exponential working vaca-

tions. When the system becomes empty, the server takes a working vacation, during which

customers are served at processing rate 7 (7 > O), If the system is empty at the end of

the working vacation, the server takes another working vacation. On the other hand, if a

customer is being served at the end of the working vacation, the server switches its process-ing rate to one and continues to serve customers in a preemptive-resume manner, until the

system becomes empty. In what fo11ows, we call time intervals during which customers aJre

served at processing rate one normal service periods. We assume that lengths of working

vacations are independent and identically distributed (i.i.d.) according to an exponential

distribution with parameter n (n > O). Let V denote a random variable representing the

length of a randomly chosen working vacation.

There are K classes of customers, labeled one to K. Let rc denote {1, 2, . . . , K}. During

working vacation periods (resp, normal service periods), class k (k E K ) customers arrive

according to a Poisson process at rate ){wv,k (resp, ANp,k). Let Awv and ANp denote the total

arrival rates during working vacation periods and during normal service periods, respectively.

Awv =EAwv,k, ANp = £ ANp,k,

kcrc kErc

where we assume Awv > O to avoid trivialities, The amounts of service requirements of

class k (k E IC) customers who arrive during working vacation periods (resp. normal service

periods) are assumed to be i.i,d. according to a general distribution function Hwv,fo(x) (resp.HNp,k(x)). For each h (ic E rc), ]et Hwv,k (resp. HNp,k) denote a random variable represent-

ing the amount of the service requirement of a randomly chosen class k customer arriving

in working vacation periods (resp. normal service periods). We denote the Laplace-Stieltjes

transforms (LSTs) of Hwv,le and HNp,k (k E rc) by hVvv,k(s) and hNp,k(s), respectively. Let

Hwv (resp. H}gp) denote a random variable representing the amount of the service require-

ment breught by a customer randomly chosen among those arriving in working vacation

periods (resp. normal service periods). We then define hVvv(s) and hNp(s) as the LSTs of

Hwv and HNp, respectively.

h"vv(s) =

il,2. A)X,",."'k

h"vv,k(s)7 hfJp(s) ==

il,l.l kill.P;k

hNp,ic(s)

We define psw,le and pNp,k (k E ]C) as pwv,ic = Awv,kE[Hwv,h] and pNp,k = ANp,kE[HNp,k],respectively. Let pwv = EkEre pwv,k and pNp = ZhErc pNp,ic.

'In what follows, we assurne

pNp < 1. [I]he service discipline is assumed-to be FIFO, unless otherwise mentioned, and

services are nonpreemptive.

Remark 1. When n > O, the sgystem is stable of and only of pwv < oo and pNp < 1. 7b

see this, consider the length C of an interzJal between successive starts of workeng vacations.

Note that the system is stable of and only of E[Cl < oo, By dofinition, C can be divided

into two parts, one of which is the length of a working vacation period Cwv with rnean 11n

and the other is the length of the foelotving normal seTwice period (iNp. Let CJeev denote the

total amount of work in system at the end of the working vacation period. if pwv < oo and

pNp < 1, the stability of the system is ensured because E[ONp] = E[URv]1(1 - pNp) and

E[o] -S+ ?[tUtsi,] gS+ ,pw-v,kn, < oo,




114 Zlnoue&T7tikine

where the fir:st inequality comes from the foct that in every sampte path, Uca is boundedabove by the total amount of work brought in the working vacation period.

Conversely, of the system is stable, E[U{iv] < oo holds, and therofbre pwv < Do. IilLr-thermore, in an ordinary M/(l/1 queue, the first passage time to the idle state with fin'iteinitial worktead is finite of and onty af the traffic intensity is less than one. Therefbre, we

have pwv < oo and pNp < 1 ij' the system is stable.

Remark 2. lf we ignore custo7ner classes, the above 7nodet is reduced to a single-ciass

MfGfl with eoponential working vacations characterixed by am'ival rates Awv and ANp,amounts of service requiTe7nents Hwp and HNp, processing rate or during worktng vacation

periods, and emponential Zengths of workeng vacation periods with mean 1!n,

3. Total Work in System

In this section, we discuss the total amount of work in system in steady state, Let U denot,e

the total amount of work in system. We define Uis･v (resp, UNp) as the conditional total

amount of work in system given the server being on working vacation (resp. being in a

normal service period). Let u'(s), u",.(s), and ukp(s) denote the LSTs of U, Uwv, and

UNp, respectively. We then have

u'(s)=jPWv･ubev(s)+IJNp･ukp(s), (1)

where Pwv (resp, INp) denotes the time-average probability of the server being on working

vacation (resp. being in a normal service period).

Let U&v denote the total amount of work in system at the end ofa working vacation. Wedenote the LST of Ueev by utw,E(s). Consider a censored workload process by removingall normal service periods. In the resulting process, the ends ef working vacations occur

according to a Poisson process with rate n. Therefore, owing to PASTA [13], we have

utw,E(s)=ulivv(s), E[UIIvlv]-E[Uwv]. (2)

We then have the fbllowing two lemmas, whose proofs are given in Appendices A and B,respectively.

Lemma 1. ukp(s) is given by

1-utw(s) ,

UNp(S)=

,E[uK..] 'uM!G!i(s)i

(3)

where uM!G!i(s) denotes the LST of the a7nount of work in syste7n in an ordinary M!G!1

queue and it is given by

(1 - pNp)s

UacIIGfi(S)=,.pN..+A..hN,(,)'

(4)

Lemma 2. Pwv and P5"p are given by

opE[Uw.] 1-pNp Pwv = &P = (5) 1 - pNp + epE[Uwv]

'

1 - pNp + "E[ULLvv] i

respectively.




FZIFO MfGll euezae with L-lorking Vkecations I15

With Lemma 1, u'(s) is given in terms of ul)vv(s) and E[Uwv].

u'(s) = pwv･utw(s) + jFNp･i,rErui'V,w"(i)･uK{iGfiCs)i (6)

where Pwv and jFNp are given in C5). We now characterize utcv(s), Note here that the conditional total amount Uwv of work

in system is equivalent to that in the corresponding M/Gll queue with Poisson disasters

[3,15]. Therefore we can readily obtain uSvv(s) using the results in [3,151. Note that asimilar observation with respect to the queue Iength has been made in [5] for a single-class

M/G/1 queue with exponential working vacations.

Lemma 3. uW.(s) and E[Uwv] are given by

"Wv(S) = ,-

A,wh(ill bV,w)S/-

ty)rh/,,..7 (,) - n!or,

E[u..] = Pwvep

-

orV,

(7)

respectively, wher ℃ y denotes the conditional steady state probability that the ser"ver is busy

given that it is on werking vacation. IVote that u is given by

U= (i(l 7) K).".Wi ni (8)

wheTe r (1' > O? denotes the untque real root qf the following equation.

z- h"vv (op/'y +Awvl'y - (Awv/ rf)z), lzb < 1･ (9)

The proof of Lemma 3 is given in Appendix C.

Remark 3 (Remark 2.2 in [15]). 71he sotution r of (op represents the probability that a

randomly chosen busy period starting in a woTking vacation ends within the working vacation.

7-b see this, consider an M/G!1 queue with arrival rate Awv, the LST hVvv(s) of servicerequirements of customers, and the processing rate 7. The LSTe*(s) of the lengths of busyperiods is then given by e'(s) = h"..(sh + Awvh

- (Awvh)e*(s)), Comparing this with

(9,), we have r = e'(n) > O,

Rearranging terms on the right side of uWv(s) in (7) yields 1-u

"WV(S)

==

1-..ftx,.c,), (10)

where f"Vv(s) is given by

jkkv(S) = (eru!Awv){n/"r ll!XYti(l)'-r ll (Awv/-y)r - s}' (ii)

Remark 4. Theorem 2 in f31 shows that .f(kv(s)

represents the LST of the reTnaining serwice

requi7ement Fwv of a randomly chosen customer present in working vacation periods when

custorners are served on a LIFO preemptive resume basis. Note that (op and (1op impty

E[iii..]=1-U,E[u,..]=1-U Pwv-7y, (12) v u ep




116 l Inoue & T 7Zikine

Theorem 1. u'(s) zs gtven by

xu'

Cs)

=

utw

(s)

LI kvv + INp1 - fWv(s)

sE[Fwv]nK(fG!i(s)) , (13)

where ufufG!,(s), u"vv(s), ftw(s), and E[Fix-v] are given by (4?, (Z), (11?, and (12?, respec-

tivety, and Rvv and 1Np are given by

I]Wv =1- pNlp

-

+ Pp'wPv

-w, jF5cp =1- KY\+ -

p.r.Y-or., (14)

respectively.

Proof It fo11ows from (10) and (12) that

i,ME7uW,V,'

fS])=: ,-'i.. .

U

(,) i,-

.f[ i",IX(i)

-tttw(s) i,-

.'[ iX,,IJ::(i),

(is)

Substituting (15) into (6) yields (13). Further (14) fbllows from (5) and (7), Z

Remark 5. Theorem 1 shows that U is stochastically decomposed into two independent

nonnegative random variables, i.e., U = Cfwv + Ui, where the LST of non-negative randorn

variable Ul is given by

uxs)=pwv+i]Np i,-Ef[tr.8)

uivGii(s)

4. Waiting Time and Sojourn Time

In this section, we consider the actual waiting time and sojourn time distributions of class

k (k E 1(;) customers in steady state, assuming the FIFO service discipline. Let wr (k c IC)denote the waiting time of a randomly chosen class k eustomer. For eaeh k (k E J℃), we

define Wwv,h (resp. WNp,k) as the waiting time of a randomly chosen class k customer

arriving in a working vacation period (resp, a normal service period). Let wX,(s), u)"iv,k(s),

and wftp,le(s) (k E IC) denote the LSTs of Wk, VVwv,k, and WNp,k, respectively. Similarly,let (?le (k E J(J) denote the stationary sojourn time of class k customers. R)r each k (k E KJ),we define (?wv,k (resp, ([l}Np,ic) as the sojourn time of a randomly chosen class k customerarriving in a working vacation period (resp, a normal service period), Let qX(s), qixfv,k(s),

(k E rc) denote the LSTs of ([l}k, Qwv,k, and C?Np,k, respectively.and qkp,k(s)

Fbr each ic (ic c rc), we define Pikv,k (resp. FISp,k) as the probability that a randomly

chosen class k customer finds the server being on working vacation (resp. being in a normalservice period) upon arrival. By definition, wn(s) and qn(s) (k E K]) are given by

w#(s)-Rl}.,,･wW.,,(s)+Il(?t,,,-wN.,,(s), (16) qX(s);Rj}v,k-qtw,k(s)+ecp,k･qkp,k(s), (i7)

respectively. Because class k customers arrive according to a Poisson process with rate

Awv,k during working vacation periods and rate ANp,k during normal service periods, 1]lj>v,kand I Iep.k satisfy

R5>v,k Awv,kPwv

]Fllilp,k ANp,kl]Np '




FIFO MfGll eue"e with Vforking Vtications 117

Therefbre, using PiiVv,k + Pl<"p,ic = 1, we obtain

Pfu'k = A..,k2i'tli'k+RAVi,,,,FN, = A..,,(1 -ApW.V.')k (t A-.P.N,,P()p.. - .y.)' (18)

Ili}p,k = Awv,k,pAwNvP'k+INA:p,k&p = ),wv,k(l iNpPN'::Ell[lxlN-

p,7("p)w. - -ru)' (19)

Both Wk and Qte (k E rc) are considered as the processing time of a certain amount

of work. More specifically, Wk (k c rc) corresponds to the stationary processing time ofwork in system seen by an arriving customer of class k. On the other hand, ([?k (k E j(;)

corresponds to the stationary processing time of the sum of work in system seen by an

arriving customer of class k and his!her service requirement, To treat Wk and Qk in aunified way, we define Twv(Ux) (resp. [INp(Ux)) as the processing time of the amount U>cof work conditioned that the server is on working vacation (resp. in a normal service period)when its processing starts, where U)c is assumed to be a nonnegative Tandom variable

whose distribution function and LST are given by C,(x(x) and uX(s), respectively. Because

the processing rate in Twv([(k) may change from 7 to one, we divide Twv(Ux) into two

parts, TiSe'l(U]sc) and [I-Sll(Ux), where TS'R,) (Ul)c) (resp. 7"IV(U)y)) is defined as the length

of a subinterval in 7Wv(Clx), during which the processing rate is equal to cr (resp. one).

By definition, Twv(C(x) = TiSJ'lr) (U>c) + TGR' ,) (UX), where T51'n,) (Uly) > O fbr Ul\ > O, and

TGI'V(Ul\)2 O. We then define ipiGv(w,s1 U>c) and ipNp(s Ux) as

iptw' (tJ,s1C(x) == E [e-ur71SJe(UX)erS71XV(Ux)], ipkp(s]U)c) = E [e-STN'(UX)],

respectively.

Lemma 4. ip"C'l.(ev,s1 C(x) and diNp(s t(x) are given by

ip"c,}.(,,,,t u.) - .i (Lv l; n) . \,X*

iS)){7.U

i' ,() il,l;-"

,)}, ipN,(,iu),) = .}(,),

respectively.

Proojl We first consider ip&p(s 1 Ul\), When the processing of U)( starts in a normal

service period, the processing rate is fixed to one throughout its processing. We then have

7Np(Ux) = Ux/1, from which ipNp(s 1 Ul\) = uN(s) fbllows. On the other hand, when the

processing of U)c starts in a working vacation period, we have

(7Viv) (uX), 7Wl(Ux)) =

where i,g denotes the remaining length of th

Owing to the memoryless property of the ex

tributed with parameter n, We then have

(C(x,o), va . !I, 7 7

(%, ux - ,va), % ( k, 7

e working vacation when the processing starts.

ponential distribution, I/k is exponentially dis-

iptw' (Lv,s 1 Ux) = fOO ciux(x) [e-"(xh) . e-w(xh)

Jo L




118 X Inoue & T Tlrkine

+ {i - e-n(xh) } L"!T

ine-n'dT- e-ny(xfny)e-WTe-SCX-IT)] ,(20)

from which the expression of dibl'v(co,s Ux) fo11ows, D

Using Lemma 4, E[TiS3'tt,} (Ux)], E[TW'l,) (U)()], and E[[INp([J>c)] are obtained to be

E [Tmu') (U)c)] = (-i)-Li.m, E:.T [ipVe'v(ev,o1 ux)] - i r

"Xn(n!7),

(2i)

E [TM') (ux)] - (-i)･2i.n, £,J[ip"G.(o,s cly)] - E[u.] -7･i-Uk(n17),

(22)

E[7Np(Clx)]=E[Ux]･ (23)

We now turn our attention to the waiting time distribution. Consider the censored

process obtained by removing all normal service periods. In the resulting process, class iccustomers arrive according to a Poisson process, Owing to PASTA, the conditional amount

of work in system seen by a randomly chosen class k customer arriving in a working vacation

period has the same distribution as Uwv. Therefbre the conditional waiting time distribu-tions are identical among classes. Similarly, the conditional amount of work in system seen

by class k (k c rc) customers arriving in normal service periods has the same distributionas UNp. Thus, the eonditional waiting time distributions are also identical amollg classes.

Let LViStll,) (resp. MrTWtV) ) denote the length of an interval cluring which a randomly chosen

customer waits for hislher service in a working vacation period (resp, normal service period),given that the customer arrived in the working vacation period, By definition, Wwv,k =

VVSItiV + WiSltV for all k (k E KJ). Also, let VVNp denote the conditional waiting time of a

randomly chosen customer given that the custorner arrives in a normal service period. Wethen define wtw' (cv,s) as the joint LST E[exp(-cvM(S['R,) )exp(-sMIGIiV)] of VViSl'IV and WGI'l,and wNp(s) as the LST of M/rNp.

Theorem 2. w""'v(w,s) and wN,(s) are given by

.tw･ (.,,)=.tw(wIny).U#v(,S)){7.".W'V,)SW

i,"

}), .N,(,)-.k.(,),

respectivety.

Proof By definition, wtw' (Ld,s) =

gbliv'.(cv,s F Uwv) and iL,Np(s) = diNp(s 1 UNp), so thatTheorem2immediately follows from Lemma 4. D

Because Wwv,k = TViktl + W5Rt r) and lttfNp,h = WNp fbr all k (k c IC),

wtw,k(s) = uiVe"v(s,s), wkp,ic(s) = w&p(s), Vk c JC.

Thus the LST u]X(s) (k E J<)) of the waiting time distribution of class k customers is obtainedby (16). In particular, the mean waiting time is given by

E[wr] - p5>.,, [(1 -

or)(1

'

n"tw(ny/or)) + E[Uwv]] + llSp,k ' E[UNp];




FIFO MIG/I Queue with H/brking V2ications

where E[UNp] denotes the mean amount of conditional work in system given thebeing in a normal service period and it is obtained from (3) and Lemma 3.

E[uN,] - Pw",

-

or + ix:?[{lil;:]) + g?iE-[l;/

2

.p)]

119

system

Next we consider the sojourn time distribution. Fbr each k (k E rc), let ([2(iti)v,k (resp.QWt)v,le) denote the length of time during which a randomly chosen class k customer spends

in a working vacation period (resp. a normal service period), given that the customer arrives

in the working vacation peried. By definition, Q(2ti)v,k > O, Q{lti)v,k ) O, and qwv,h = Q("t)v,ic +

Q(1'l).,,. We define q"C'i.,,(w,s) (k c K]) as the joint LST E[exp(-w([?M') ,,)

exp(-sc? £k'l).,,)] of

(?<2'iV,k and ([2Sti')v,k, and qkp,k(s) (k E K]) as the LST of QNp,h, 'Theorem 3. qtw'

,k(w,s) and qftp,k(s) (k E JC? are given bgy

q"e'.,k(ev,s) =u"v. (W I; n) h",v,ic (w;n) uW.(s)

･ hW.,,(s) - uW.

+ (wlln) h",.,,(wIIn)

(or!n){(w + n)h -

s}7

qNp,k(s) -= uNp(s)

･ hftp,k(s)i

respectivelgy.

Proof By definitien, qW'v,k(w,s) = iptw' (w,s 1 Uwv + Hwv,k), and qNp,ic(s) = ipNp(sCINp+HNp,k). Theorem3then fo11ows from Lemma 4. a

Note that q",.,k(s) - qtw' ,k(s,s)

(k E rc). Thus the LST qX(s) (k E IC) of the sojourn

time distribution of class k customers is obtained by (17). In particular, the mean sojourn

time is given by

E[([?,] = R51}.,, ･ E[([?..,,] + PISI,,, ･ E[([2Np,k],

where

E[(E}..,k] -

E[([?.,,kl -

(i - or)Ci - uWv(nh)htw,le iv/or)) + E[[fwv] + E[Hwv,k]; nE[UNp] + E[HN,,,].

5. Joint Distribution of Queue Lengths and Work in System

In this section, we consider the joint distribution of the numbers of customers and the

amounts of work in system in respective classes, rlb

do so, we first derive the joint LST

of the attained waiting time and the rernaining amount of service requirement of a class kcustomer being served, With this result, the joint distributions are derived.

For each k (k c rc), let aQtl,k (resp. aX)v,k) denote the timeaverage probability that class

k customers, who arrived in working vacation periods, are being served in working vacation

periods (resp. in normal service periods). Also, let akiP,h (k E )(J) denote the time-average

prebability that class k customers arriving in normal service periods are being served.

NII-Electionic Libiaiy




120 llnoue&T7lakine

Lemma 5. aWt<,) ,k,

aec)v,h, and akiP,k (k E rc? are gzven by

a£{'iV,k=K'v " "lll(:tk:}Zl･liliS,(7,/?)))･

(2`)

aee'v) ,,-p.. [p..,,-7u Al::.\;kElIZi.

f･(ki/"or!;)))], (2s)

(7gliP,,=IiN,･p.,,,, (26)respectively.

The proof of Lemma 5 is given in Appendix D.Remark 6. Let a denote the utiltzation foctor, i.e., the time-average probabiZity that cus-

tomef:s are being served. Recall that u in (8) represents the conditional probability of theserver being busy gtven that the server is on working vacation. VVe then have

(1 - pNp)u + pwv -

crfv ff =: 1- Fwv'(1-y)=Rvv 'u+ 1]Np= ･ 1 - pNp + pwv

- rfu

Rtrthermore, using Lemma 5, we can verof21

2 {7g'Rf) ,k = Pwv ･ u, 2(aS'i)v,k + aSliB.k) - iiNp･

kErc kErc

We now consider the attained waiting time [8], which is defined as the length of time

spent by a customer being served (if any) in the system. When the system is empty, theattained waiting time is defined to be zero. Note that under the FIFO service discipline, all

waiting customers in the system arrived during the attained waiting time.

For later use, we divide the attained waiting time into two parts: One is the (sub)intervalin working vacation periods and the other is the (sub)interval in normal service periods, LetA£1'k)

,k

(k E J(]) denote the length of time in the attained waiting time, cluring which theserver was on working vacation, given that a class k customer is being served. Furthermore,fbr each k (k E 1(]), let A £ltl)v,fo (resp. Aki3,k) denote the length of time in the attained

waiting time, during which the server worked in a normal service period, given that a elass

k customer, who arrived in a working vacation period (resp, a normal service period), isbeing served. Fbr a class k (k c rc) customer being served, let Hic denote the remaining

amount of hisfher service requirement. We then define the following joint LSTs:

a"G'v,wv,k(cdk,ak) = E[eLWkAW'Vke-akfite ZtCBarSoScekssCiUnSgtOrlllteer"S being

Served],

a"V'{r,Np,k(cvk,sk,ctric)=Ele'wleA £"'Vlee-skAW')vke-akfik i.CkallSl.frk:･.UgStvOaMceart'ioW.hpOe?lordileidt 1, L being served at processing rate one]

aN'p,k(slt, cik) = E [e"skAkiP ke-Qkfik

ft.C,ll:S.Sl k,,C,".i::rBgr,l,.Wdh,

1,abrrelV.egd,:.aed]See Figures 1-4, where Figure 1 corresponds to aV"'v,wv,k (wk, cvk), Figures 2 and 3 correspondto ae('7{f' ,Np,k(tuk,sk,ak), and Figure 4 corresponds to ak'p,k(sk, cik).

Moreover, for each k (k E rc), let U&tl,k (resp. HWtlf) ,k)

denote the lengths of time duringwhich a class k customer, who started hislher serviee in a werking vacation period, is served



The OpeiationsReseaichSociety ofJapan

FIFO M/GII e"eue with VVlorking liZications 121

- Ume

arrival

"""

Figure 1: Attained waiting time of a class k customer in a working vacation period

during the working vacation period (resp. the subsequent normal service period), We then

define fitw' ,k(w,s)

as the joint LST of H5J'l,) ,k

and Hant) ,h.

Using Lemma 4, we obtain

A"e･v･k(w･s)

e.e

lw

P'

le

J)S"

l'L

iS/I,;ip

;,ll,:,2X.t･S

i/,,(llil/lln)

We then have the fbllowing theorem, whose proof is provided in Appendix E.

Theorem 4. a"('tv,wv,te(Lvh, clic), a"e'<f' ,Np,k(cvic, sic,aib), and aN'p,fo(sk, cek) are given by

atw･ ..,(w,,a,) (

.

i

i'

,,

)

.r.,iYi!

W

,.

'

S,e? h""

g;(

,

a

/;

)

(.

-

,//,

v

);,(i

k

.or',}

"),

(,,)

a"e'V' ,Np7h(LVk, Sk,ak) =

E [([?gtl)v,kl-

vvlisltv,h] [u (CVk +

n)

ftVe'V･k(CVk)

dvskk)

-- ch,

"

trevk(Wki Sk)

uevv(sh)nu

hw.,k(ak)-hWv,k(sk)

t (or!n){(wh+n)fty-sk} s,-a, , (28)

hNp,k(ah) -

hNp,k(sk)

tw (.k7+ op)

aN'p,ic(sk,ak)=uNp(sk)'

E[Hs,,,,](,,-.,) i (29)

respectively, where E[([?<I'l,) ,k

- PVSI'il,kl and E[(?(-'a,) ,,

- WrGR' ,) ,,]

are g2ven in aZ) and (4i!?,7espectively,

With Theorems 2, 3, and 4, we can verify that aVV'v,wv,k(wk, crk) and aV}'IJ' ,Np,ic(Wk,

Sk, dvk)

are represented in terms of wCV'v(w, s) and qV('/v,k(w, s)-

Corollary 1. atw' ,wv,ic(wk,crk)

and a"if'<i' ,Np,is(wk,sk,cvle)

are gzven by

aec･,.,..,,(w,,

dvic)

-

WW']f

£ kta

,

k.)

21i,[r･i,(?

`

.

)

.,f,) ,,

-

-q",llV' j,lie.lf,i'

cric)

･



The Opeiations Reseaich

l22

Society of Japan

ZInoue&T

Tlakine

- working vacation perio

Figure 2: Attained waiting time of a class

working vacation period and will end his!her

d normal service period

k customer who started his!her service in a

service in a normal service period

Figureperiod

working vacation period nQrmal service period

3: Attained waiting time of a class k customer who

and started his!her service in a normal service period

Figure 4i

arrlved ina working

Attained waiting time of a class k customer who arrivedin

vacation

a normal service period




FIFO MfGli e"e"e with lllorking 1ikecations

a"e' {r' ,NP,k

(CVki Sk, ak)

(ZVtw' (Wk,Sh) - tVtw' (Wk,dvh))htw,k(dvh) - (qtw' ,k(Wic,Sk)

- q"e'v,k(Wk,ak))

123

(sk-ak)E [([?(1'l)v,k-WiS3'l) ,k]

'

respectively,

Let Lwv,k (resp, LNp,h) (k C )C) denote the number of class k customers in the system,

who arrived during working vacation periods (resp, normal service periods), Also, let Uwv,fo

(resp. UNp,k) (k E 1(J) denote the amount of work in system, which is brought by class k

customers who arrived during working vacation periods (resp. normal service periods). We

then define the joint transfbrm tb(zwv , zNp, swv , sNp) as

ip(Zwv, ZNp, Swv, sNp) = E [" (.7.wiWvV,kk ･ zkP?rdk ･ e-Swv-kOinrv k . e-sNp kONp,k)1 ,

Liccrc J

Where Zwv = (Zwv,1,ZiW,2?･･･iZWV,K), ZNP = (ZNP,1iZNP,2,･･-,ZNP,K), SWV = (SWV,1,SWV,2,..･,SWV,K)) and SNP = (8NP,1,SNP,27･･･,SNP,K)･Theorem 5. ip(zwv,zNp, swv, sNp) is given by

V(ZWV,ZNP,SWV?SNP)

= (1 - v) Piw + 2 2wv,ka £ijtV,kaV"'v,wv,ic ( 2 [Awv,z - Xwv,zxwv,2htw,,(swv,t)] , sxw,h)

kErc iEre

+ E zwv,kaeetlr) ,katw'*

,Np,k(2

[Awv,, - Awv,izwv,thtw,,(swv,i)] ,

leCre iErc

Z [ANp,, - ANp,,zNp,ihNp,,(sNp,D] , slw,k)

iErc

+ E iNp,foakiB,hak'p,lt (2 [ANp,, - ANp,tzNp,thkp,,(sNp,z)] , sNp,k) ･

kErc iErc

Proof1 Note first that the system is empty with probability 1 - a = (1 - v)IMrv (seeRemark 6). Furthermore, when a customer is being served, all waiting customers arrived

during the attained waiting time, as noted at the beginning of this section. Theorem 5

immediately follows from those observations. a

Remark 7. Let Lwv (resp. LNp? denote the total number of customers in the systern, who

arrived during working vacation periods (i'esp. normat ser'vice periods?. Atso, let U/wv (resp.Utsrp? denote the total amount of worben syste7n, which was brought by custorners who arrived

during woricing vacation periods besp. normal seTn]ice periods?. As stated in Remark 2, we

can obtain those by considering the single-class 3ystem zvith Awv, htw(s), ANp, and hNp(s),

77ierofbre Theo7e7n 5 also provides the formula for the j'oint tranofbrm of Lwv, LNp, C,rwv,

and CINp implicitly, because it corresponds the case ofK == 1.

Taking the partial derivatives of ip(zwv, zNp, swv, sNp), we can obtain the moments of

Lwv,k, LNp,k, Uwv,k, and UNp,k (k E )C). In particular, we have

E[Lwv,kl = Awv,kPiw ' E[([?wv,fo], E[LNp,k] = ANp,k&p ' E[([?Np,h]i

E[0wv,k] = pwvpwv,h (E[uiNv] + 2EE[iZ/iff

'

IX'?l] + ;) - X (a£['k) ,k + aSl'i)v,k) i

E[ON.,,1 = IiN,p.,,, (2EE[ltlft.P:\1 + E[uN,I)




124 l lno"e & T 7lrkine

6. BusyCycle

The busy cycle is defined as the interval between ends of successive busy periods. In order

t･o analyze the busy cycle and related quantities, we first consider the first passage time to

the empty system. More specifically, we define ]Fkiv-v (resp. FIFip) as the first passage t･imeto the empty system given that the server is on working vacation (resp, in a normal service

period) at time O. We divide Fwv into two parts: F51tle) and FQtl, where F"(iiv) (resp. ,Fl51iv))

denotes the length of a subinterval during which the server is on werking vacation (resp.in a normal service period). By definition, Fwv = FStiV + Fl5ei,v) . Furthermore, for each k

(k c rc), we define NS(tR,) ,k

(resp. NXv) ,ic)

as the number of class k customers arriving in F5(tR,)

(resp. .Fl&i-V ), Similarly, we define AINp,ic (k E i(J) as the number ofclass k customers arriving

in FNp. Let S(t) (t 2 O) denote the state of the server at time t, i.e., and S(t) = WV if theserver is on werking vacation at･ time t, and otherwise S(t) = NP. We define U(t) (t 2 O)as the total amount of unfinished work at time t. We are interested in the fo11owing joillttransforms,

g<XTv(zwv,zNp,swv,sNp x)

=E[(kfiLrczwNetJSkk zNNtW,irk) e"swvFWV ersNpF;iJV u(o)=x

gSlp(zNp,sNp E x) =E [(icRLrczNiVpNPkle) e-sNpFNp u(o) =x,s(o) =Np]

Where Zwv = (Zwv,1, Zwv,2, ･ . . , Zsw,K) and ZNp = (gNp,1, ZNp,2, ･･･ , ZNp,K)･

Lemma 6. <Np(zNp,$Np x) is gtven by

<i3p(ZNp, sNp 1 m) = e-fiRp(zNp7sNp)x,

where 6Np(zNp, sNp) is dqfined as

s(o) - wvl ,

]

1

C30)

6Np(zNp7 sNp) =

sNp + ANp - ll,Ilrc zNp,icANp,h ][

Oe

<kp(zNp, sNp 1 y)dHNp,k(y), (3i)

and it is given by

ffNp(zNp, sNp) = s., + A., - E) 2..,icA.p,hhkp,k (6k,(z.p, s.,)). (32) kErc

The proof of Lemma 6 is given in Appendix F.

Next, we consider the joint transform C"v.v(zwv,zNp,swv,sNp x), Given S(O) = WV,let 7-V denote the time instant when the server ends the current working vacation for the firsttime after time O, Because of the memoryless property, [TV･ is exponentially distributed with

parameter n. We classify the first passage time Fwv to the empty system into two cases,

avv g 7]v and Fwv > 7V, and we define <ikv,c(zwv,swv x) and Ctw,E(zwv,swv,a I x)as

Ctw,c(zwv, swv 1 =) = E [(kHLrc zwNet[ltick) e-sw.F}Xre

ccVV,E(ZWV]SWVia X)

u(o) = ., s(o) = wv, F.. s [u,l ,

1



TheOpeiations Reseaich Society of Japan

EIFO MIGfl Queue with Lllorking li2ications 125

= E [(hn,rc zXlj'&kk) e-SwvlV ･ e-aU(TV) u(o) = m, s(o) = wv, avv > [Iv] ,

respectively. Note here that (30) implies

E [(,H,.z.i".W,is'le) e-SNPFtlV U(O) =x,S(o) =wv,Fwv > 7i,,u(7v) =: y]

= E [ (kn, xJ zNNpN ,Pk･k) eJSNp

FNp

u(o) = y, s(o) = Np]

= Ci}p(zNp,sNp y) = e-Skp(zNp,sNp)y

We then have

Cfuv(ZIW,ZNP,SWV,SNP X)

= klx ' CWv,c(zwv, swv 1 x) + Pklx ' C{Iiv,E(zwv, swv, 6ilrp(zNp, sNp) 1 x), (33)

where I]er. and Ilid. are defined as

i]bl. -= Pr(Fwv S [l-V I U(O) - x,S(O) -

WV),

lk]1. - PrCFwv > [IV U(O) ::: x,S(O) -WV).

Lemma 7. T7ie following eguations hold.

klx'CWv,c(Zwv,Swrvix)=e-ffWV,C(ZWV'SWV)X, (34) e-aX - e-SWv,c(ZWv,Swv)=

･6"vvE(zwv,sww,cM), (35)

I]kl.･C"vv,E(ZwviSWViaPX)]= 6tw,c(zwv,swv)-a '

where 6"vv,c(zww,swv) and fi"iv,E(zwv,swv,a) are dofined as

ep Awv SWV

6tw,.(z..,s..) - +-+ 7 or 7

- ll,ll.

ZXW'k7XWV'k

ygOO

R 1,･ctw,.(z,w,s..1y)dHww,k(y), (36)

fitw,E(zwv, swv, a) = n/7

+ ll,lilrc ZW"iicorXWV'k

y(iOO

I]be, ･ Ctw,E(zwv, swv,a y)dHwv,ic(y), (37)

and they sattofZt

6tw,c(zwv,swv) =

S\llY

+ ll + )LlllV

- ll,ll.

Zwy'k.rAWV'k

･ htw,k(ptw,c(zwv,swv)), (3s)

BGvv,E(zwv,sww,a) ll+ii,i,,lgWV'k.yAW""ic

hVVV'h(aiiSl.iil(.<kt(lii:.\ig/(Z-W."iSW"))

'fleev,E(zwv,swv,a). (39)




126 l Inoue & T Tlakine

The proof of Lemma 7 is given in Appendix G.

It fbllows from (33), (34), and (35) that CIIvv(zwv,zNp,swv,sNp x) is given by

<iXrv(ZWvi ZNp, Swv, sNp 1 :) = e-S"iv,c(Zwv,Swv)x

e-6Np(XNp,SNp}= L e-S"iv.cCZwv,Swv)=

+ 6tw,c(Zwv,swv) - 6K,.(zNp,sNp)

' I3tw7E(XWViSwv,6Np(zNp,sNp)). (4o)

With (36) and (37), we define 6wv,c and Pwv,E as

P..,c = rs<V.,c(1,0), 6wv,E = ffW.,.(1,O, 0),

where 1 denotes a vector whose elements are all equal to one.

Lemma 8. Pwv,c and 5wv,E are given by

nh eWV,C=6wv,E=1-., (41)

and R l. and I]bl. are given by

R]1.=e-fiWV'C'=, R]1.=1-e-6W"･C'=, (42)

The proof of Lemma 8 is given in Appendix H.

We now consider the busy cycle. Recall that the server is always on working vacation at

the beginning of busy cycie. Let e denote the length of a randomly ¢ hosen busy cycle. Wedivide e into two parts, and let e(7) (resp. e(i)) denote the length Qf the subinterval duringwhich the server is on working vacation (resp, in a normal service period), Furthermore,we divide e(or) into two parts, and let eX) (resp. ek')) denote the length of the subinterval

during which the server is idle (resp, busy). By definition, e = ek7) +eET) +eCi). For each

k (k c rc), let N)E7) (resp. NEi)) denote the number of class k customers arriving during e(i)(resp, O{i)). We then define the joint transform of those quantities as fo11ows.

e'(Zwv7iNp,w)swv,sNp) == E[(kllLrezecEv']ic xNN-31) e=blekD e-swvek'} e-sNpea)l

By definition, e'(zwr, zNp, cd, swv, sNp) satisfies

e'(zwv, zNp, cv, swv, sNp)

=

. Ii\l.l . Il,llrc ZWVskwXvWV

k

y[oo

ctw(zwv,zNp, swv,sNp y)dHwv,k(y)

Therefbre, with (4e), we obtain the fbllowing theorem.Theorem 6. e*(zwv,zNp,co,swv,sNp) is given by

o*(zwv,zNp, Lv, swv, sNp)

;

av ll)lftVwv llLl)lc ZWVAkwAvWV,k

htw,k(I3N,(zNp, s..)) - hO,.,k(l3tw,c(zwv, swv))

+ Btw,c(zwv,swv) - 6"p(zNp,sNp)

r

[htw,k(eWv,c(zwv,swv)) 1･

6tsr,E(zwv

,

slw

,

fikp(zNp,

sNp))1





FIFO M!G/l eaeue with Tllorking V2ications 127

Remark 8. It is clearfrorn the derivation of Theorem 6 that

Pr(A randomly chosen busy pentod ends zvhile the server is on working vacation)

=

Li.m, . I)lill.. Il,ll. AAW.".'k

'

htw,h (6"vv,c(o, o)) =

htw(fiwv,c) =

r,

where we use (5ep, This result is consistent with Remark ge. thrthermore, usMg (lf?Erp and

(99), u;e obtain an alternative e:;pression fbr e'(zww, zNp, Ld, swv, SNp)･

e'(zwv, zNp,w, swv, sNp) = w l\wv [Ak {swv + Awv

- orBtw,c(zwv, swv)

+ arfrstw,E (Zwv , swv , 6ftp(zNp , sNp)) }] Taking the partial derivatives of e'(zwv,zNp,w, swv, sNp), we can obtain the moments

of NET), NS), eEor), and e(i). In particular,

E[N27)] = Awv,k (Ak + E[ek7)]) , E[N £i)]

= A..,k - E[e{i)1,

E[ek7)]-7nX3AttC-Ak., E[e(i)]=(1-r)･El[U-WpV.]iU,

7. ConcludingRemarks

We considered the stationary multi-class FIFO M/G!1 queue with exponential working

vacations. We derived the LST of the stationary work in system, and the LSTs of the

stationary waiting time and sojourn time in each class. We also obtained the joint transform

for the queue lengths and the amounts of work in system in respective classes and the

joint transfbrm associated with the busy cycle. Before closing this paper, we provide some

remarks.

As stated in section 1, if we delete time intervals in nermal service periods from the

time axis, the resulting process can be viewed as a multi-class FIFO M!Gll queue with

Poisson disasters, where the processing rate is equal to ty. Because queues with disasters

are of independent interest, Appendix I summarizes the analytical results fbr the multi-class

FIFO M/G/1 queue with Poisson disasters, all of which are immediately obtained from the

results in this paper.

In queueing models with working vacatiens, the processing rate is always equal to 7 when

the system becomes empty, In other words, the queue length process directly affects the

processing rate. From this point of view, the queueing model with working vacations differs

from the queueing model embedded in a random environment (i.e., the processing rate is

assumed to change according to an underlying envirenmelltal piocess). More specifically,

we can see the diffk)rence between these two models by considering a special case of our

model, where the processing rate is proportional to the arrival rate and comparing it to the

corresponding queue embedded in a random environment of a two-state Markov chain. In

the latter, the stationary number of customers in the system is independent of the underlying

Markov chain and its conditional distribution given a specific state of the Markov chain is

the same as that of the ordinary M/G!1 queue (Section6 in [11]). 0n the other hand, it is

verified that the model we considered does not have such a property. Thus, the queueingmodel with working vacation is essentially different from the queueing model embeclded in

a random environmental process.




128 Z lnoue & Z 1lakine

AppendixA. Proof of Lemma 1

We define Ul?p as the total amount of work in system at the beginning of a normal service

period. Note that URp is a conditional random variable of Uca given that the server is busyat the end of a working vacation. Let uNp,B(s) denote the LST of Uillp. We then have

tts,,,.(s) :=E[ers"{lv u(kTv>o] =U"V"iE iSl;il(cPJ.ii.(t-'"oi-

O), (43)

E [UIIip] = i- pE,IUu'Ie

E

V.]=

o) (")

Consider a censored workload prQcess by removing all working vacation periods from the

time axis. In steady state, the censored process has the same distribution as UNp. Also, thecensored process can be viewed as the conditional workload process of the M!G!1 vacation

queue with exhaustive services, given that the server is busy. Therefore, it fo11ows from (5.6)in [2] that uNp(s) is given by

.k.(,) = 1IEUi:ill?(]S)

uK,!.1,(s)

Note here that (2), (43), and (44) imply

1-uSlp,B(s) 1-utw,E(s) 1-uVvv(s)

sE [Ui?p] sE [Ueev] sE[Uisrv] '

which completes the proof.

B. Proof of Lemma 2

We regard an interval between successive ends of working vacations as a cycle. Let Cwv

(resp. (:INp) denote the length of an interval during which the server is on working vacation

Cresp. in a normal service period) in a randomly chosen cycle. Owing to the renewal reward

theorem, we ha;ve

&rv = E[c.?.[]0+WVEIcN,] ) iiNp = E[c.E.][CiiliZi][on,]

Because Cwv is equivalent to the working vacation length V, we have E[Cwv] == E[V].the other hand, E[Cbgp] equals to the mean first passage time to the empty system in

corresponding ordinary MfG!1 queue with initial workload of Cli?p. Noting that CNp =

the system is empty at the end of the working vacation, we have

E[Ul?p] E[Uwv]

E[cNpl

=

pr(U{Bv

=

O) ･ 0+

{i - Pr(Ul

i'v =

O)}

' i - p.,

=

i - pNp'

where we use (2) and (44). (5) now follows from (45), (46), and E[Cwv] - E[V] = 11".

(45)

Ontheo

if

(46)

C. ProofofLemma 3

The censored process obtained by removing all normal service periods is considered as an

M/G/1 queue with Poisson disasters with rate op, where the system becomes empty when





FIFO MIG!1 eueue with Tlhrking Vlrcations 129

disasters occur. The M/G/1 queue with Poisson disasters has already been studied in [3, 15],where the processing rate is assumed to be one, In order to apply the results in [3,15] to

our system, we consider the new process created by extending the time axis of the workload

process in working vacation periods or times so that the processing rate becomes one. Notethat the time-average quantities of the new censored process are identical to those of the

original process. In the new process, the arrivaJ rate of customers is･equal to Awvh and

lengths of working vacations are exponentially distributed with parameter vrh. utw(s) in

(7) then immediately fo11ows frorn Proposition 1 in [3]. We also obtain (8) by substituting O

to the repair time in (2.la) in [15]. The existence of the unique real root of (9) is shown in

Remark 2.2 in [15]. See Remark 3 for the positivity of r. Furthermore, taking the derivativeef utw(s) in (7) and evaluating at s = O yields E[Uwv] in (7).

D. Proof of Lemma 5

We first consider aWtV,k. Note that all customers being served in working vacation peri-

ods arrived during working vacation periods. Thus, from Little's larw, we have a£"tl,k =

Awv,kPwv ･ E[QE"'L,, - WiS('ll], Furthermore, with Lemma 3 and (21), E[9<"').,, - Wi"'l] is

obtained to be

E [([2£I'lf) ,k - WISJ'e] = E [TSR' r) (Uwv + Hwv h)] - E [7RS(tylv) (Uwv)]

= u"vv(n/'>')(i

; hVvv,k(ny/">'))

= Aii. ･ ii--hhW'

'

elV:{(nn!/./S) 7 (47)

from which (24) follows.

Similarly, ffW'lv) ,h

fo11ows from ael'i}v,k = Awv,kkv ･ E[([2W')v,k - VViSl'V] and

'

E[Qwh-wQ'V]I/kL T

tliii'

cil-wv

,;...

"W,V

,Lh-

'lti.t(i,ll/III,,l(UWV']

,,,,

Finally, we consider gkiB,k. Note that all customers arriving in normal service periods

are served in normal service periods, Therefore ase,k = ANp,leIiNp ' E[HNp,k] = i]Np ' pNp,k,from which (26) fo11ows.

E. Proof of Theorem 4

We first consider (27). Suppose a class k (k c rc) customer is being served at processingrate cr (i.e., in a working vacation period). Note here that

E[c?(1,R,) ,,

- wX'l,, Cl}W').,, - W51'ie,, > Ol -E[c2<2'l).,,

- W5('Yl,) ,,]

uWv(ep/or)

We thus have

aVe'v,wv,ic(wle,aic)

1u"v (wk +n7)

E[(?W,Y,, - WiS('RJ) ,k]uWv(nh)

utw(n!or)



The Opeiations Reseaich

130

Society of Japan

X Inoue & Z 7lakine

. ./[

OO

dHwr,k(x) [e-n(=h) y[X!i

eTwkte-ak{xrTt)dt + y[=h

ne-nTdT L'

e-wkte-ak(=-Ait)dt] ,

from which (27) fo11ows.

Next we consider (28). Suppose a class k (k c )C) customer, who arrived in a working

vacation period, is being served at proeessing rate one (i.e., in a normal serviee period). Wethen have

E[c2w' ,) ,, - wix';v,, i cE}g';'.,, - vv:sll,k > o] - , -E.ivq,.£1'iiV,'lihl(･Illlliil'

,tk,]/,)･

Therefore

a"e'V,NP,k (LVk, Sk, ak)

1

E[Q£k'2v) .k - WW'lr)

,k]

i - utw(nh)hWv,k(n!7)

[uivl(l.I.7),;i,e,Zl.uiuti)･"" i.,IW

,i7

'

,,

")

' i - hwli,k(nj.r) Ygoo

dHwv,k(x) Y[=h

ne-nTdT f'+"-cr'

e-wkTe-sk(t-T)e-a,(x-v"(t-.))dt

1-u",.(nh) 1 n"vv(sk)-u",. (Wkor+

n)

+ . ,

i - uWv (nlty)htw,k(n!or) (or/n){(wk + ny)/7 -

sk} 1 -

uWv (n!or)

y[OO

dffwv,k(x) L=

e-3k`e-ak(X-`)dt] ,

from which (28) fo11ows.

Finally, a&*p,le(sk, ak) is given by

1 aN'p,k(sk, ak) =

E[HNp,k]

from which (29) fbllows.

F. Proof of Lemma 6

For x ) O, y) O, we have <i3p(zNp,sNp I x+y) =

Therefbre

CNp(zNp, sNp I x + Ax) = gS}p(zNp, sNp 1 x) ･ CS3p(zNp

= gkp(zNp,sNp

)LNP,k

ANp

･ uNp,k(s) y(]

De

dHNp,k(m) yg=

e-skte-crk(x-t)dt,

gkp(zNp,sNp 1 x) - CNp(zNp, sNp L y)･

,sNp1Ax)

: x) [i - sNp[ix - ANpAx + ANpAx llLllrc xNp,k

y[eOCk,(zNp,sNp

y)dHNp,k(y)+o(Zlx)],





FIFO MIG/I eueue with T{/brking 1/2ications 131

from which it fo11ows that

8. [<ftp(z.p, s.p Px)] = -CSI.(zNp, sN.1x)fiS},(zNp, sNp).

Noting <i3p(zNp,sNp O) = 1, we obtain (30). Also, substituting (30) into (31), we obtain

(32).

G. Proof of Lemma 7

It is easy to see that fbr x ) O, y 2 O,

R lx+y ･ C{Xrv,c(zwv,swv x+y) = R]lx ･ CWv,c(zwv,swv 1 x)

･ Pbly ･ <Wv,c(zwv,swv 1 y),

Il]1.+y･Ctw,E(zwv,swv, cM1x + y) = ]Fki.･gtw,E(zwv,swv,a1x)-e-orY

+klx･Ctw,c(zwv,swvlx)'lkJy'GWv,E(zwviSwv,a1y)･

Therefbre we have

R]1=+Ax ･ Ctw,c(zwv, swv 1 x + Ax)

==Pbl.･Ctw,c(zwv,swv)x)･aPblA.･Ctw,c(iwv,swv Ax)

= kix ' <tw,c(zwv, swu a x) [i - swy lt - r] IX - )Lwv A.yX

+Awv liiX llLilK

2WVsicwAvWV'k

yleO

kf, ･ gu,v,c(zwv,s,w y)dHwv,h(y) +o(Ax)] ,

from which it fo11ows that

oO. [i]br=･gtw,c(zwv,swvFx)] = -Pbg.･("vv,c(zwv,swv1x)･fitw,c(zwv,swv)･ (49)

(34) now fellows from (49) with ]Fblo ･ CIXrv,c(zwv, swv 1 O) = 1･

Similarly, FkEx+Ax ' <Wv,E(zwv , swv , a x + Ax)

= Fblx'<ikv,E(zwviswv,cM1x)-e-orAx

+ke= ' <l)vv,c(zwv7swv 1 x)

' jFklAx ' GXrv,E(zwv,swv,or Ax)

= Pblx ' gcav,E(zwv, swv,a 1 x) ･ (1 - aAx) + o(Ax)

+ R]ix ' CiXiv,c(zwv, swv 1 x) [n A7X

+ A.. A7X ll,ll.

ZWVsfowA.WV7ic

y[OO

Iikl, t c{;vv,E(zw., s..,a y)dH..,k(y) +o(Ax)] ,

and therefore

2il.r [iiL]i.<tw,E(zwv, sww , a I x)]

= -all]s. ･ CWv,E(zwv,swv,a x) + R]lx ･ Ctw,c(zwv,swv m)fiWv,E(zwv,swv,a)･ (50)

Multiplying both sides of (50) by eaX and using (34) yield

Slt [l]blxG(Xrv,E(zwv, swv, a 1 m) - eaX] = e'fi"'v,c(ZWVTSWV)X - fitw,E(zwv, swv, a) ･ ea=.





l32 ZInoue&T7Z]kine

Because lklo =: O, we obtain

IIkl. - <tw,E(zwv, swv, cv l x) ･ eax ; y[

=

e-{S"iv c(zwv･swv}-or}y

x3tw,E(zwv, swv, cv)ctgl,

from which (35) fbllows. Substituting (34) into (36) yields (38), and substituting (35) into(37) yields (39),

H. Proof of Lemma 8

(42) follows from CWv,c(1,O L m) = 1, Il]1. + Pkl. = 1, and (34). We thus consider (41)below. Note that Ctw,E(1,0,Olx) = 1. Therefore, taking the limits a

-> O and s --> O in

(35), we obtain

]Fbix =

:l:\ill ･ (i - e-Swv,c'=),

from which and (42), we have 6wv,c = 6wv,E･ It is readily seen from (38) that 6wv,c satisfies

13wv,c == n/'r+Awvl"r-(Awv/or)hWv(rswv,c), (51)and h"vv(fiwv,c) = htw (olor + Awvh - (Awv/or)h",.(ffwv,c)). Furthermore, we have from

(36)

6wv,c =

nh + Aw.1or - ,2..(A..,,for)

Y[OO

Pbl,dH..,,(y)

= eph+ Awvlor - (Awvh) y[

OO

I]bl,dHwv(y) ) n/or > o, (s2)so that lhVvv(6wv,c)F < 1. As a result, htw(6wv,c) is identical to the minimum nonnegative

root r of (9), Finally, from (8) and (51), we obtain

nh ep!7+Aiwh-()Lwv/:t)r=1-., (53)which completes the proof.

I. The Multi-Class FIFO MIG/1 Queue with Poisson Disasters

In this Appendix, we summarize the results ofthe stationary multi-class FIFO M!G/1 queuewith Poisson disasters, where the processing rate is equal to one. We can readily obtain

those results by ¢ onsidering the conditional counterparts in the multi-class FIFO MIG/1with exponential working vacations and or = 1, given that the server is on working vacation.

I.1. Model

Consider a stationary multi-class FIFO MIGfl queue with Poisson disasters. Class k (k EK]) customers arrive according to a Poisson process with rate Ak. Let hk(x) and hX(s)

(k E J(]) denote the distribution function of service times "k of class k customers and itsLST, respeetively. Disasters occurs according to a Poisson process with rate n (n > O), andthe system becomes empty when disasters occur. We define A and h(m) as

A-2Ak, h'(s) -2 lit' ･ hx(s) kErc kEK]

Note that if we ignore customer classes, the system can be regarded as a single-class FIFOM/G/1 queue with Poisson disasters. Note also that the system is stable regardless of values

of system parameters.



The OpeiationsReseaichSociety ofJapan

F7FO MfGIJ Queue with V-lorking Vhcations

IiNW = Pr(UA g DA),

2llk(s) = E[e'S" j UA S DA],

and for each k (k E K)

Pi5ilk = Pr(UA + Hk g DA),

qN,,(s) - E[e-SQk 1 UA + "k f{l DA],

By definition, we have

w*(s) = E[e-SL"] = I]NWwk(s) + I]bWzvb(s),

Because LV corresponds to UtrQ71 in

Theorem 2

w'(s) =

Note here that u'(s + n) = I]NWu;N(s).

(56) represents I]bWw6(s).

.N(,) = u'Sdi)n) ,

4We=u*(ep), 4Ws

Similarly, it follows from Theorem 3 that

1 qX(s) u*(s+ny)hX(s+n)+

133

I.2. Results

The LST u*(s) of the amount of work in system is given by [3,15] (cf. Lemma 3 and its

proof)

ttx(s) - , -(k i :)hS* (

-,)ny-

n

Note that y denotes the stationary probability of the server being busy.

u= (i (l ;) K)ln' (s4)

where r denotes the minimum nonnegative root of the fbllowing equation.

z= h'(op +A- ){z), lzl <L C55)

We denote the amount of work in system seen by a randomly chosen customer on arrival

by U]-,, and the length of the interval from the arrival of this customer to the occurrence of

the next disaster by DA. Owing to the memoryless property, DA is exponentially distributedwith parameter n. We define wr and Qk (k c rc) as the waiting time and sojourn time,

respectively, of class k customers, i,e., Wk = min(UA,DA) and ([2k = min(UA + Hh,DA)･Note that ewing to PASTA, LVk (k E rc) is identical to the waiting time PV of a randemly

chosen customer, Furthermore, we define

l]bW - Pr(UA > DA),

wb(s) - E[e'SW 1 UA > DA],

,FtiSlk ==

qB(s)

q£(s)

Pr([Lx + Hk > DA),

= E[e'SQk UA + Hk

= E[e-SQk] = IIEII,k

> DA].

qft,k(s) + Ii{il,k

the queue with working vacations,

u'(s + ny) + lif,"i((,S .' ,"))

Therefore the second term on the ri

It then fo11ows that

1 1-u'(s+n)

WS(S)

1-u*(n) (1/n)(s+n)' - 1 - u' (op).

- u"(s + op)hZ(s + n),

qb,k(s).

we obtain from

(56)

ght hand side of

(1in)(s + ny)k E ]C,




134 Z Inoue & T. 7lakine

and therefore fbr each k (k E JC)

qft,k(s) -

U'(S.:(Zik((X)'

") ･ qb,k(s) = , - ., (i,),z(,)

' i -

"ii'xZ)3ii

' ") i

il[ilk =- u'(ep)hZ(n), 49h -i-u*(")hn(n)･

Let ak (k E rc) denote the probability of a class k customer being served, which corre-

sponds to a £['k) ,k/kv

in the queue with working vacations. It fbllows from (24) that

Ak(1 - hZ(ep))

ak

="'

A(1- h*(n)) '

where y is given in (54). Let Ak (k E rc) denote the conditional attained waiting time given that a class k customer

is being served and Iet Hk (k E rc) denote the remaining service time of class ic customerbeing served. We then define aZ'(sk,ak) (k C rc) as

at*(sk,ak) = E[e-SkAk ･ e-akllk l a class k customer is being served],

Note that aX(sk,ak) corresponds to a"e'v,wv,le(wk,ah) in the queue with working vacations.

Mereover, Qic and Wk eorresponds to Q£ltR,) ,h

and WXV,le, respectively. It then fo11ows from

(27) that

ax*(sk,ak) - E"[

'

Q(Ik-'iili)i] ･ hn(a,k,)

i ,hXiS.le,' ") ,

where E[Qk - VVkl is obtained from (47).

E[Q, - wk] -: K･ li Zl:Zi .

Let Lk (k E K]) denote the number of class k customers in the syst･em and Iet Uic (k ( ]C)denote the total amount of work in systern belonging to class k. We then define the jointtransform ab(z, s) as

tb(ZiS) =E [k"LrcxkLk e-SkUk] ,

where z = (zi, z2, . . . , zK) and s = (si, s2, . . . , sK). We then have

cb (z, s) = 1 - u + 2 zkaic aX' (2 [Ai - At zi h,* (s,)]7 sk) ;

kCK iErc

which corresponds te Theorem 5.

Finally, we consider the busy cycle, which is defined as the interval between successiveends of busy periods. Let e denote the length of a randomly chosen busy cycle. We dividee into two parts, and Iet eE (resp. eB) denote the length of the subinterval during whichthe server is idle (resp, busy), We define M (k E )C) as the number of class k customersarriving during e. Let UL denote the amount of work in system that is lost due to disasters.We then define joint transforms eN(z,s) and eS(z,s,a) as fo11ows.

eft (z, cv, s) == E [(,ll,re ZNk) e-weE -seE･e 1a

busy

period

ends

without

disastersl

,



TheOpeiations Reseaich Society of Japan

FIFO MfGll Que"e with Illorking 1,Zications 135

OS(z, cv, s, ci) = E [(kllLrc zNle) e'WeE ･ e-SeB ･ e'aOL a busy period ends with disasters]

We also define Il? as

Ill? == Pr(a busy period ends without disasters),

and let Pl? = 1 - Fl?. It then fbllows from Lemma 7 that

il? t eft(z, Lv, s)

=

A l ,, ll,2. xk - lit' y[

OO

e-fift(z･s)ydHk(y)

=Al ,, 2xk･

Akhn (X3ff*

(Z' S))

(s7) kErc

A s+ep+A-ffk(z,s)

(58) - t

r)L+cv A

'

il? ･ ek(z,w, s, a) = Al.

k2,rc zk

･ lit' y[

OO

e-

6

a

.l i.,e,

'

)

61(Z.'S)Y

･ s6(z,s, a)dHk(y)

= A lw li,i,i, Zh ' AAle

' hZ(Ctz/NII.4Z,()flij

(.Z; S))

' sb(zis, a)

A 6b(z,s,a)-ny (59) - -

rA+w A

'

where 6&(z,s) and 6S(z,s,a) satisfy

6N (z, s) ; s + n + A - £ zkAkhZ(f3R (z, s)), kEre

e6 (z, s, a) =

n+ ll,li. zkAk - hA("2sc?:X,()fig

(i S))

･ eb (z,s, a),

which correspond to x3tw,c(zwv, sNw) in (36) and Ptw,E(zwv, swv, cy) in (37), respectively.

We define 6N and fiD as

/3. - 6il,(1,O), e. - P{,(1,O, O).

We thell hewe (cf. Lemma 8 and its proof)

6N = /3D = i IZ ., h'(fiN) = r,

where r is the minimum nonnegative root of (55). It then fo11ows from (57) that

ill? =r) IS =1- ri

and from (58) and (59) that

A 6S (z, s, dv) - ep A s+ep+A-S&(z,s)

ek(z,u),s)=A+,,･

ny+A-/3. , eb(zitv7s7a)=A+,.

p.-op ･

NII-Electionic



The Opeiations Reseaich Society of Japan

136 ZInoue&T. Tbkine

References

[1] O,J, Boxma and T, Takine: The M/G/1 FIFO queue with several customer classes.

queueing Systems, 45 (2003), 185-189.

[2] B. Doshi: Conditional and unconditional distributions for )vl!G/1 type queues with

server vacations. queueing Systems, 7 (1990), 229-252.

[3] G, Jain and K, Sigmani A Pollaczek-Khintchine formula for MfGll queues with dis-

asters, Joumal of Applied ProbabiZity, 33 (1996), 1191-1200.

[4] J. Kei!son and L.D. Servi: A distributional fbrm of Little's law. Operations Research

Letters, 7 (1988), 223-227.

[5] J,D. Kim, D.W, Choi, and K.C. Chae: Analysis of queue-length distribution of the

M/G!1 queue with working vacations (M!G/1!WV). Proceedings of Hawaii inteTna- tional Conjerence on Statistics and Related Fields (2003), 1191-1200,

[6] J.-H. Li, N.-S. Tian, A.G. Zhang, and H.P. Luh: Analysis of the MfG/1 queue with

exponentially working vacations A matrix analytic approach. eueueing Systerns, 61

(2009), 139-166.

[71 W.-Y. Liu, X.-L. Xu, and N,-S. Tian: Stochastic decompositions in the M!M!1 queue

with working vacations. Operations Resea7=h Letters, 35 (2007), 595-60e,

[8] B. Sengupta: An invariance relationship for the G!Gfl queue. Advances in Applied

Probability, 21 (1989), 956-957.

[91 L.D. Servi and S.G. Finn: M/M/1 queues with werking vacatiens (M/M/11WV), Per-

formance Evatuation, 50 (2002), 41-52.

[10] T. Takine: Queue length distribution in a FIFO single-sever queue with multipte arrival

streams having different service time distributions. 9ueueing Systems, 39 (20el), 349- 375.

[11] T. Takine: Single-server queues with Markov-modulated arrivals and service speecl.

9ueueing Systems, 49 (2005), 7-22.

[12] T. Takine and T. Hasegawa: A generalization of the decomposition property in the

M/G!1 queue with server vacations. Operations Research Letters, 12 (1992), 97-99.

[13] R.W. Wolff: Poisson arrivals see time averages. Operations Research, 30 (1982), 223-

231.

[141 D.-A. Wu and H. Takagi: MfGll queue with multiple working vacations. PeTfbrmance

Evaluation, 63 (2006), 654-681.

[15] W.S. Yhng, J.D. Kim, and K.C. Chae: Analysis of M/G!1 stochastic clearing systems.

Stochastic Analysis and Applications, 20 (2002), 1083-1100.

Ybshiaki Inoue

Graduate School of Engineering

Osaka University

2-1 Yamadaoka, Suita Osaka 565-0871, Japan

E-mail: [email protected],jp

In [5,6,14], - J-Stage

Documents