行政院國家科學委員會專題研究計畫 成果報告
具遞增損害率之時間特性的流程分析模式
計畫類別： 個別型計畫
計畫編號： NSC93-2213-E-004-006-
執行期間： 93 年 08 月 01 日至 94 年 07 月 31 日
執行單位： 國立政治大學應用數學學系
計畫主持人： 陸行
報告類型： 精簡報告
處理方式： 本計畫可公開查詢
中 華 民 國 94 年 10 月 3 日
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An Analytical Model for Service Systems with Increasing Failure Rate Processing Times
: NSC 93-2213-E-004-006 : 930801940731 : Æ
1 Abstract
Our goalis to investigate theoutput process under whatconditionstheinterdeparturetimewillpreserve theIFR property. Because ofthe complexity ofthe stationary probabilitydensity, wetake advantageof computerto visualizetheperformanceoftheoutput process. We found the interdeparture time doesn't alwayspreservetheIFRpropertyevenifthe interar-rival time and service time are Erlang distribution-s with IFR. We give several theoretic analysis and present somenumericalresultsof E
m =E
k
=1queues. Forourexperiment,ifmk,theinterdeparturetime ofE
m =E
k
=1remainstheIFRproperty.
(Keywords: DepartureProcess,PH=G=1,Hazard rate,IFR.) i_lm PH=G=1 Æ ÆLST !"#$% &Æ'() Æ*+,-./01./20345 & ÆIFR67 8Æ9:;<=> ?@ABIFR36C9DEFG ÆHI6 JCKLMNOPE m =E k =1Æ !QR@0,ST UVWXYZ [\]^+_BIFR67@`a bABIFR36cd Æef UV g9E m =E k =1 hmi9jk9k^@ ABIFR67
(nop:qLr%,PH=G=1,Hazardrate,IFR.)
2 Introduction
Fordecades, queueingnetworks hasbeen basic ana-lyticmodel forthestudy ofcomputers, communica-tion networks and manufacturing systems. Recent-ly, the study of the networks shifted their focus to thedepartureprocessesinthequeueingsystems, be-cause the departure processes from one server may be the input or arrival processes of another server. Therefore,it isimportanttocharacterizethe depar-tureprocessesin thequeueingsystems.
From thepass studies, we knowa GI=G=1 queue hasoutput instants which form arenewalprocessif and only if the arrival process is Poisson and the services times are exponential. When the output processis notarenewalprocess, researchersstudied thecorrelationstructureoftheoutputprocess. This is because departure processesof queuesother than M=G=1 are diÆcult to characterize. In this paper, ourgoal is to investigate what propertieswill be p-reservedforthedepartureprocess ifthequeueisno more M=M=1 queue. We will consider a PH=G=1 queueand construct theLaplace-Stieltjes transform (LST)oftheinterdeparturetimedistribution,where
Neuts [11, 13, 14, 15]. The phase typedistribution canbeagoodapproximationforgeneraldistribution. Weanalyzethestochasticproperties,suchas Increas-ing Failure Rate (IFR) for the interdeparture time. BecauseofthecomputationalcomplexityofPH=G=1 andthestationaryprobabilitydensityofthenumber of customers in the system, werestrict our numeri-calexamplestotheE
m =E
k
=1queue,whereboththe interarrivaltimeandservicetimeareErlang distribu-tions. Weillustratesomenumericalresultsand con-siderunder what conditionstheinterdeparturetime isIFRin theE
m =E
k
=1queue.
Sincethe1950's,manyresearchers[1,3,7,8]have studiedtheoutputprocessofqueueingmodels. They focusedonthedistributionoftheinterdeparturetime and the correlationstructure of the output process. Inthispaper,weuseanalternativeapproachto an-alyze the performance of the output process. We consider the stochastic order relations of the inter-departuretime,suchasIFR.Theyareimportantfor comparisonof\new"and\residual"lifetimes[2,4]. Thestochasticorderrelationsoftheoutput process-esfrom oneservermaybeimportantindices forthe inputprocessesofanotherserverinthenetwork[5,6]. In this study, we investigate under what conditions theinterdeparturetimewillpreservetheIFR proper-ty. Wetakeadvantageofcomputertoconduct some experimentsand visualize thefailure rate of the in-terdeparturetimeof E m =E k =1queues 3 Performance analysis
Now,weconsidersomeimportantperformance mea-suresoftheinterdeparturetimeforE
m =E
k
=1queue.
Lemma3.1 ForthestationaryE m =E k =1queue,we have E[D]= m X j=1 0 (j)(m+1 j) m + 1 :
Lemma3.2 ForthestationaryE m
=E k
=1queue,we have the variance of the interdeparturetime Var[D]
m X j=1 0 (j)(m+1 j) 2 (m) 2 + m X j=1 0 (j)(m+1 j) (m) 2 ( m X j=1 0 (j)(m+1 j) m ) 2 + 1 k 2 :
Lemma3.3 ForthestationaryE m =E k =1queue,we haveE[D]= 1 which implies m X j=1 0 (j)(m+1 j)=m(1 ):
Theorem 3.4 For a stationary E m
=E k
=1 model, we have the square coeÆcient of variation of the interdeparturetimec 2 D = Var[D] E 2 [D] 1and c 2 D = 1 m 2 m X j=1 0 (j)(m+1 j) 2 + 1 m (1 ) (1 ) 2 + 1 k 2
Byadi erentapproach, Buzacottand Shanthiku-mar[2]showedthatforanyGI=G=1queuewith DM-RL(decreasing mean residual life)interarrival time, theupperbound onc
2 D is c 2 D c 2 A (1 )+ 2 c 2 S + (1 ), and whenthe interarrivaltime have IMR-L(increasing mean residual life) property, the lower bound on c 2 D is c 2 D c 2 A (1 )+ 2 c 2 S +(1 ), where c 2 A and c 2 S
are the the square coeÆcient of variation of the interarrival time and service time, respectively. Since Erlang distributions are DMR-L and hyper-exponential distributions are IMRL, it is easy to check with the upper bound condition of the E
m =E
k
=1 queue that has c 2 D
1. With the lower bound condition, it is easy to show that the H m =H k =1queuehasc 2 D
1inwhichtheinterarrival timeandservicetimearebothhyper-exponential dis-tributionswithc 2 A andc 2 S
beinggreaterthanone.
4 Stochastic properties
Inthissubsection,weconsiderthefailureratefor sta-tionaryinterdeparturetime ofE
m =E
k
preservetheproperty ofIFR evenifthe interarrival timeandtheservicetimearebothIFR.Wewill con-siderunder what conditionstheinterdeparturetime preservethepropertyofIFRfortheE
m =E
k
=1queue. weemploythemethodofpartialfractionsthat sep-arates itin terms of and respectively[9,10, 12]. Wehave ~ D(s)= m X j=1 a j ( m s+m ) j + k X i=1 b i ( k s+k ) i ; (1) wherea j andb i
arecoeÆcientsassociatedwitheach termofand. Wewilldiscussthemethodofpartial fractions and how to obtaina
j and b
i
in numerical examples.
Takingtheinversetransformofaboveequation,we havethedensityfunction oftheinterdeparturetime isintheform of d(x)= m X j=1 a j (m) j (j 1)! x j 1 exp( mx) (2) + k X i=1 b i (k) i (i 1)! x i 1 exp( kx): (3) ThecoeÆcientsa j andb i
areattainedinaccordance to the method of partial fractions and the station-ary probability
0
that a departure leaves the sys-tem empty with respectto the arrivalphases. First we consider the initial valueof the failure rate r(x) of theinterdeparture time distributionand we have thefollowingtheorem.
Theorem4.1 For the stationary E m
=E k
=1 queue, the initial value of the failure rate r(x) of the inter-departuretimedistributionis
lim x!0 + r(x)= ( (1 0 e) if k=1 0 if k2: where 0
is the stationary probability that a depar-tureleavesthe systememptywith respecttodi erent arrivalphases.
Nowweconsider the nalvalueofthefailurerate r(x) of the interdeparture time distribution. Since
convergesto asx ! 1, where = k, does the failure rate of the interdeparture time distribution r(x)convergeasx!1andwhat isthelimit? Theorem 4.2 For the stationary E
m =E
k
=1 queue, the nal valueof the failure rate r(x) of the interde-parture timedistributionisgiven by
lim x!1 r(x)= ( m if mk; (km) k if m>k; (k<m): Thatis lim x!1 r(x)=minfm;kg:
5 Hazard rate analysis of E
m
=D=1 queues
Inthissubsection,weconsideradeterministicservice case. Since the Erlang-k distribution converges to a constant value as k ! 1. According to Section 2,wecanobtainseveralperformanceindices forthe departure process. Nowweexaminewhether ornot the interdeparturetime of E
m
=D=1queue has non-decreasingfailure rate. ByTheorem ??, the failure rater(x)ofthestationaryE
m
=D=1queueisgivenby (?? )whereh=1=andr
I
()isthefailurerateofthe idle timedistribution I(). The LST ofI()is given by ~ I(s)= P m j=1 0 (j)( m s+m ) m+1 j +(1 0 e). Let i(x) betheprobabilitydensityfunction ofI(x). We have lim x!0 + I(x)=1 0 e (4) and lim x!0 + i(x)= 0 (m)m: (5) Sincer I (x)= i(x) 1 I(x) ,wehave lim x!0 + r I (x)= 0 (m)m 0 e : (6)
Form=1,namely,M=D=1queue,wehaveI(x)= 1 (1 )exp( x)forx>0. Then,lim
x!0 +r
I (x)=
havelim x!h
+r(x)<=(1 ). Thisimpliesthatthe output process forM=D=1 queue doesn't have IFR property. Ifm2,wehavelim x!h + r(x)=lim x!0 + r I (x)= 0(m)m 0 e
. We will discuss whether lim x!h +r(x) is larger than (1 0 e)= 0 e or not by numerical methodin nextsection.
6 Conclusions
Inthisproject,wederivedtheLaplace-Stieltjes trans-form (LST) of the interdeparture time of PH=G=1 queue and gave some indices for the performance analysisof thedepartureprocessofPH=G=1queue, suchasthemoments,thevariance,andthesquare co-eÆcientofvariation. WeshowedtheE
m =E k =1queue hasc 2 D 1andtheH m =H k =1queuehasc 2 D 1. Especially, we analyzed the failure rate of the s-tationary interdeparture time. To the best of our knowledge, ithasnotbeenstudied beforeinthis as-pect. We focused on the IFR property of the in-terdeparture time of E
m =E
k
=1 queue. Because of the complexity of the stationary probability densi-ty
0
, we took advantage of computer to visualize the performance of the output process. We found the interdeparturetime doesn't alwayspreservethe IFR property even if the interarrival time and ser-vice time are Erlang distributions with IFR. But if km,theinterdeparturetimeofE
m =E
k
=1remains theIFRpropertyinourexperiments.
References
1. Burke, P. J.(1956), The output of a queueing system,OperationsResearch4,699-704.
2. Buzacott, J. A. and Shanthikumar J.G.(1993), Stochastic models of manufacturing systems, Prentice-Hall.
3. Daley, D. J.(1976), Queueingoutput processes, Advancesin AppliedProbability8,395-415.
Markov-modulated Poisson process (MMPP) cookbook,PerformanceEvaluation18,149-171. 5. Ishikawa,A.(1991), Onthejointdistributionof thedeparture intervalsin an M=G=1=N queue, Journal of the Operations Research Society of Japan34,422-435.
6. Jenkins, J. H.(1966), On the correlation struc-ture of the departure process of the M=E
=1 queue, Journalof the Royal Society B28, 336-344.
7. King, R. A.(1971), The covariance structure of thedepartureprocessfromM=G=1queueswith nitewaitingline,JournaloftheRoyal Statisti-calSocietyB33, 401-405.
8. Laslett, G. M.(1975), Characterizing the nite capacity GI=M=1 queue with renewal output, ManagementSciences22,106-110.
9. Lee, Y. H. (2002), The Output Process of E
m =E
k
=1,Masterthesis,NCCU.
10. Luh,H.(1999),DerivationoftheN-step interde-parture time distribution in GI=G=1 queueing systems, European Journal of Operational Re-search118,194-212.
11. Neuts,M.F.(1989),Structuredstochastic matri-cesof M=G=1 typeandtheirapplications, New York: MarcelDekker.
12. Osaki,S.(1992),Appliedstochasticsystem mod-eling,Springer-Verlag.
13. Yeh,P.C.andJin-FuChang(2000), Characteriz-ingthedepartureprocessofasingleserverqueue from the embedded Markov renewalprocess at departures,QueueingSystems35,381-395. 14. Saito, H.(1990), The departure process of an
N=G=1 queue,PerformanceEvaluation11, 241-251.
15. Takagi, H. and Nishi, T.(1998), Correlation of interdeparturetimesin M=G=1and M=G=1=K queues,JournaloftheOperationsResearch So-cietyofJapan 41,142-151.
