Analysis of Restricted Queueing Networks - A Blocking Approach

Transcription

1 Journal of tatstcal cence and Applcaton, Octber 06, Vol., No. 09-0, 0-0 do: 0.765/8-X/ D DAV I D PUBIHING Analyss of Restrcted Queueng Networks - A Blockng Approach M.. reekala and M. Manoharan Department of tatstcs Unversty of Calcut Kerala (Inda) In ths paper we consder an open restrcted ueueng network wth Posson arrvals and exponental servce tmes at statons and present an approxmaton method for analyss, made by node-by-node decomposton employng the concept of effectve arrval rate and effectve servce rate. The steady state characterstcs of the network system and of ndvdual statons are obtaned. Congeston due to blockng has been a severe problem n several health care studes, but among studes that apply ueueng theory, those analyzng the blockng mechansm are stll lmted. Ths study apples to a health care system of nfertlty clnc consdered as an open ueueng network wth blockng caused due to the capacty of the watng space between statons. The method s llustrated wth the help of an example network to study the effcency of the approxmaton method. Fnally a smulaton study s performed to assess the transent behavour of the network system. The blockng approach presented here for a smple model of a healthcare system can be easly adapted to any general open restrcted ueueng network. Key words: ueueng network, blockng, steady state, performance measures, fnte capacty systems.. Introducton Queueng networks are generally classfed as restrcted ueueng networks or unrestrcted ueueng networks. Unrestrcted ueueng networks are those networks where all the ueue nodes have unlmted capacty, whle restrcted ueueng networks have fnte capacty at some or all the ueue nodes. Dependng on the flow n the system, ueueng networks can be further categorzed as open unrestrcted, closed unrestrcted, open restrcted and closed restrcted. The capacty restrcton of a ueueng network s a crucal aspect n ts study and ndeed there are many physcal systems havng fnte capacty. In manufacturng systems, computer systems, communcaton networks, healthcare systems, supermarkets, arlne termnals, and almost all physcal envronment, there are lmtatons on the capacty of the servce provded as well as the accommodatons of the customers. Majorty of the research on ueueng networks has concentrated on unrestrcted networks due to ther computatonal tractablty and well behaved mathematcal characterstcs. In restrcted ueueng networks, "blockng" may occur due to restrctons on each ueue length. The flow of customers from a node wll be blocked f the correspondng destnaton node s full. Blockng n such a networks wll be handled on the blockng mechansm that s beng adopted. A blockng mechansm s a set of rules that determnes the blockng and

2 Analyss of Restrcted Queueng Networks - A Blockng Approach unblockng of a node. Blockng occurs when watng spaces between statons are fnte. Hunt [7] examned a fnte buffer between two statons whch s an exact soluton method and computatons are more complex. Hller and Bolng [6], Perros and Altok [], Takahash et al. [] consdered smlar stuatons and suggested dfferent approxmaton methods. Korpororal et al. [] analyzed the congeston level n pentetary n Netherlands. Kozum et al. [0] uses decomposton algorthm n mental health. Health care publcatons nvolvng analyss of blockng are rare. El-Darz et al. [] and Cohen et al. [] have used smulaton studes and Hershey et al. [5] and ess and McClan [5] used mathematcal studes for the analyss of ueues wth blockng. In a recent work of Mantz [] mult-stage assembly-dsassembly ueueng networks wth blockng after servce have been analyzed usng a decomposton approach to evaluate the performance of the network. In ths paper we present an analyss of restrcted ueueng network system wth specal reference to an example stuaton of a healthcare system. For such a system we analyze the congeston n the system by applyng a ueueng network model wth blockng. Two performance ndcators, the number of patents watng to enter each type of faclty n the system and the assocated watng tme are derved n the steady state analyss. The addtonal feature of the network consdered n ths paper s n that all the feedback customers are combned and drected to the frst staton. Ths knd of feedback s prevalent and more realstc n nfertlty clncs and other health care systems. Ths study helps to reduce the unnecessary watng tme by changng the capacty as well as dentfyng the strateges to reducng ueue length.. Model descrpton e consder a health care system of nfertlty clnc as an open ueueng network system, where patents arrval at the clnc s accordng to a homogeneous Posson process and get self-servce wth a general servce tme dstrbuton for completng an ntal non-clncal regstraton formalty. Accordng to the revsed glossary of the Internatonal Commttee for Montorng Asssted Reproductve Technology (ICMART) and the orld Health Organzaton (HO) the clncal defnton of nfertlty s "a dsease of the reproductve system defned by the falure to acheve a clncal pregnancy after twelve months or more of regular unprotected sexual ntercourse" (see Zegers-Hochschld et.al [6]). For the treatment of ths dsease the male and female partners are evaluated for nfertlty or subfertlty usng a varety of clncal nterventons and also from certan laboratory evaluatons on the ndvduals. At the ntal regstraton faclty, specfc enures about lfestyle and sexual hstory of couples are collected to dentfy people who are less lkely to conceve. After completng ths formalty they enter nto the system, that s the external arrval node namely admsson node ( ) whch follows an M M FCF schedule. Here the forms are checked by the server and the people are dvded nto two categores accordng to years after marrage and readmsson. Clnc uses ths as the ntal predctor and does not check the other detals n the frst node. ess than one year after marrage ( < ) couples are routed to the second node namely counselng ( ) wth probablty α. More than one year after marrage couples are routed to the thrd node namely treatment ( ) for treatment wth probablty (- α ). In the former case, the detals of the couples are checked and they undergo a counselng process. Ths node follows an M M K FCF schedule. After the counselng sesson, the couples who realze that they don t reure treatment leave the system wth probablty α. If they want treatment, drected to the thrd node wth probablty (-α ). The thrd node also follows an M M K FCF schedule. At ths node couples are treated and they leave ths system wth probablty α. Ths type of leavng may be due to completon of

3 Analyss of Restrcted Queueng Networks - A Blockng Approach treatment (such as one day treatment) or gvng up of treatment. ome couples want more cycles of treatments and some may be mentally stressed after consultatonand want counselng, so all such feedback couples are routed to fourth node namely readmsson ( ) wth probablty (- α ). Ths node also follows M M K FCF schedule. The readmtted patents are admtted to ther respectve nodes through the admsson node. o the couples readmtted for counselng are routed to the second node wth probablty and the couples readmtted for further treatments are routed to the thrd node wth probablty ( - α )and the process contnues.. Methodologcal framework e consder an open ueueng network consstng of four sngle server nodes/statons wth fnte and nfnte capactes and the statons are desgnated by (=,,, ). Customers arrve to the system accordng to a homogeneous Posson process wth rate λ and get self servce for nonclncal regstraton wth a general servce tme dstrbuton. The customers are then entered to the frst node ( ) wth rate λ. e assume that the servce tmes are exponentally dstrbuted wth rate µ, µ, µ and µ at the respectve nodes =,,, and there s an nfnte watng space capacty at the frst node and other three nodes are of fnte capactes. A dagrammatc representaton of the above healthcare ueueng network s shown n Fgure. α Fgure. Health care ueueng network wth blockng. Blockng occurs n ths network because the watng space between nodes two ( ) and three ( ) and three ( ) and four ( ) are fnte. In ths context t may noted that the arrval to each node s accordng to Posson process wth a certan rate. hen blockng exsts, congeston at any partcular staton could affect the

4 Analyss of Restrcted Queueng Networks - A Blockng Approach congeston levels at all upstream statons. o usng the concept of effectve servce tme ( see Takahash et al. []), servce tme s modfed n order to corporate blockng between statons. Another assumpton s that all patents arrvng at any staton n the system are treated eually wthout offerng any prorty. In the model, the statons, and have fnte capacty. o blockng occurs at the flows and. Blockng wll not occur f the mmedate staton has nfnte capacty. In our model staton has nfnte capacty. o the flow s blockng free. Then the remanng flows,, are modeled. hen patents at fnd or full, they wat at staton. The Table summarzes the types of congeston assocated n ths model. Table Types of congestons assocated wth each flow Flow Cause of congeston Facng staton Congeston type to s full Classc Congeston to s full Classc Congeston to s full Blockng to s full Blockng to Not applcable Not applcable No Congeston. teady-state Analyss Assumng that the nflows are compatble wth system capactes that ensures the long run steady state behavour of the model we frst look nto the analyss of the present network ueueng model wthout consderng blockng between statons. e shall then propose the modfcaton n order to ncorporate blockng. Followng the steady state methodology for steady state analyss of ueueng models (see Gross and Harrs [] and Bose [] ) we proceed as follows.. teady-state analyss wthout blockng In our model, the routng probablty matrx s expressed as follows: R= r j, j= 0.,,,, where r j s the routng probablty from staton to staton j. Here the staton 0 represents outsde system. Thus r j s the fracton of patents departng from staton to staton j n any gven unt of tme (, j = 0,,,, ). Clearly the routng probablty matrx correspondng to the network s : α α 0 R = α 0 0 α 0. α α Based on the routng probabltes, the external arrval, the nternal and the total arrval rates of the network system t may be noted that traffc euatons (see Gross and Harrs [] and Bose []) for our system are:

5 Analyss of Restrcted Queueng Networks - A Blockng Approach λ = λ+ r λ (.) λ = r λ (.) λ = r λ + r λ (.) λ = r λ. (.) ubsttutng the routng probabltes the above euatons becomes λ = λ+ λ (.5) λ = αλ (.6) λ = ( α ) λ + ( α ) λ (.7) olvng the above set of euatons we get, λ = ( α ) λ. (.8) λ λ = α ( αα ) + αα αλ λ = α ( αα ) + αα λ( αα ) λ = α ( αα ) + αα λ λ( α )( αα ) =. α( αα ) + αα (.9) (.0) (.) (.) e know that (Jackson [8], [9]) f there s an nfnte buffer between statons n the network, each staton can be solved ndependently. o we assume that there s an nfnte buffer between statons. o each staton can be solved ndependently by applyng M M model. In a tradtonal ueueng network model lke an open Jackson network model wthout blockng, patents at a staton arrve, receve servces, and depart to the next staton. In ths system, the steady state of each staton s analyzed wthout consderng the mpact of congeston at any partcular staton on the flows at other statons n the system.. Average ueue length and Average ueue delay Expected ueue length at staton s obtaned by the followng formula

6 Analyss of Restrcted Queueng Networks - A Blockng Approach 5 ρ =, ( ρ ) (.) λ where ρ = < ( =,,, ). µ The expected steady-state watng tme to enter staton can be obtaned usng ttle s formula, =. λ Therefore = ρ λ ( ρ ), =,,,. (.). teady-state analyss wth blockng hen statons are of fnte capacty, congeston at any partcular staton could potentally affect congeston levels at all upstream statons. In ths model we modfy the Jackson approach to capture such nteractons between statons. Usng the concept of effectve servce tme Takahash et al. [] we analyze ths ueueng network wth blockng. Accordngly the effectve servce tme s a combnaton of treatment tme and blocked tme. Treatment tme represents the tme between physcal arrval to staton and conceptual arrval to staton +. Blocked tme represents the tme duraton between physcal ext from staton and conceptual arrval to staton + and that s thus euvalent to the watng tme to enter staton + from staton. In networks wthout blockng, treatment tme s eual to servce tmes. For mathematcal convenence, n our model we assume that the effectve servce tme follows exponental dstrbuton and mean effectve servce tme at staton s denoted as µ. It s the mean treatment tme ( ) µ plus the mean watng tme to enter staton + ( say + ). In general effectve servce tme s the convex combnaton of effectve watng tmes as gvenby = r ( ) +Σ r ( + ), µ µ µ 0 j j j where r 0 represents the routng probablty of patents leavng the system from th staton wthout facng any wat and r j represents the routng probablty from th staton to the j th staton. Hence n our system, statons and face blockng ( =, ). Then, the effectve servce tme correspondng to and are respectvely, = r ( ) + r ( + ), µ µ µ 0 (.5) = r ( ) + r ( + ). µ µ µ 0 (.6)

7 6 Analyss of Restrcted Queueng Networks - A Blockng Approach By substtutng the routng probabltes, the above euatons become, = α ( ) + ( α )( + ), µ µ µ (.7) = α ( ) + ( α )( + ). µ µ µ (.8) Usng the last euatons wth (.) and (.), we get the steady state mean ueue lengths and watng tmes n terms of effectve servce tmes at the respectve statons.. Analyss of tatons e analyze the steady-state of each staton usng the node-by-node decomposton approxmaton technue proposed by Takahash et al. []. In ths approxmaton technue, the network s broken down nto sngle nodes wth revsed servce and arrval parameters whch are then studed separately. o usng the sngle node decomposton method ths network s solved ndependently and ndvdually from last staton to frst staton. The steady state analyss of each fnte staton s done usng sngle node decomposton approxmaton method wth each staton follows M M ueue.. Analyss of staton four( ) In our system, staton represents the patents those who want to re enter or readmt the clnc. Those patents from reure reappearance at the node (feedback) wll be drected to. Ths s a fnte capacty node, but ts downstream node s whch has nfnte watng space. o the patents at staton do not face blockng. Thus, the effectve servce tme correspondng to staton s, =. µ µ o ueue length and ueue delay are obtaned by usng the formulas gven n (.) and (.). The staton receves patents only from staton. Therefore ueue length at the staton, whch s the steady-state number of blocked patents at watng to enter denoted by s eual to tself.. Analyss of staton three( ) In ths network, staton represents staton for treatment. There are two ways to enter staton, one drectly from admsson staton and the other from counselng staton. hen patents arrvng from staton experence blockng f the staton s full. o the ueue length and ueue delay correspondng to staton are obtaned by solvng (.) and (.) n terms of the effectve servce tme expressed n (.8). The ueue length at each staton to enter staton s obtaned by the followng euatons. and = r, (.) = r. (.)

8 Analyss of Restrcted Queueng Networks - A Blockng Approach 7 Note that s the number of blocked patents at staton watng to enter staton and s the number of blocked patents at staton watng to enter staton.. Analyss of staton two( ) In ths network staton s the staton for counselng. taton receves patents only from admsson staton, whch ncludes new arrvals and readmtted patents. Thus ueue length (.) and ueue delay (.) of n termsof effectve servce tme are calculated usng (.7). The staton receves patents only from staton. o the number of patents at watng to enter s ueue length of tself. =. (.) 5. Numercal Analyss A Matlab program has been developed for fndng the steady state soluton and the values of performance ndcators (PI). The steady state probabltes and the performance measures wth and wthout blockng are computed for the parameter values λ = 0., α = 05., α = 0., α = 06., summary of the analyss s gven n Table. Table ummary of steady state analyss 07 =. µ, = µ 0., = µ 07., µ = 0.. and the taton PI th Blockng thout blockng Results show that the ntensty of congeston s more at staton than at staton. atng tme to enter ths staton s hgh because at staton there s a downstream of two statons and. Ths stuaton may be due to the fact that the number of leavng patents s less than the number of readmtted patents. o by ncreasng the servce rate of staton, watng tme of staton s reduced and the system wll be stablzed and bounded n ths network. In fgure we plot the watng tme aganst the mean servce tme at staton. e observe that when servce rate (recprocal of mean servce tme) ncreases watng tme decreases.

9 8 Analyss of Restrcted Queueng Networks - A Blockng Approach Fgure. Congeston at staton. 6. mulaton analyss e have used the mathematcal model of the ueueng network for the steady state analyss. The transent soluton of ths complex model s uet tedous. However, one may seek a smulaton study to know the transent behavor of the network system. The same arrval rates and servce rates used for mathematcal analyss are also used n the smulaton study. Dscrete event smulaton engne "mevents" n mulnk Matlab lbrary s used for the smulaton study. e performed the smulaton study for dfferent smulaton tme vz. 00, 000, 0000, etc. The Table summarzes the typcal output of the smulaton study. Autocorrelaton result shows that all sgnal values are hghly correlated. The result of coeffcent of varaton mples that most of the sgnal values are effcent. Table ummary of smulaton results (smulaton tme 000) taton PI Average Mnmum Maxmum Autocorrelaton CV In Fgure we plot the watng tme to enter aganst the mean servce tme at the staton. It shows a comparson between the result obtaned by analytcal study and smulaton study. atng tme correspondng

10 Analyss of Restrcted Queueng Networks - A Blockng Approach 9 to varous servce tmes of staton are plotted n both the studes. It further reveals that the calculated watng tme and smulaton watng tme do not dffer too much. On further enhancng the smulaton tme we fnd that the results closely agree wth the analytcal results. Fgure. mulaton study versus mathematcal study. 7. Concluson e studed the structural aspects of restrcted open ueueng network wth blockng and the methodology s llustrated through a healthcare system of nfertlty clnc. The congeston between statons due to blockng are studed usng decomposton approach. A Matlab program was developed for calculatng the performance measures for an example network system. A smulaton study has been conducted usng mevents n Matlab program. As expected, there s dscernng dfference between the results obtaned n ether methods, however when the smulaton tme ncreases the results agree better. It may be noted that autocorrelaton and coeffcent of varaton study of the smulaton values show that the smulaton outputs are hghly correlated and more or less effcent. o from the result of smulaton study we conclude that though the mathematcal steady state results are easer for computaton, n the presence of autocorrelated sgnal values, t may not be used. Although the work presented here apples the blockng approach to only a smplfed model of a health care system, t can easly be extended to ncorporate addtonal features and appled to any open restrcted ueueng network. References []..K. Bose, An ntroducton to ueueng ystems, Kluwer academc / Plenum publshers, New York, 00. []. M.A. Cohen, J.C. Hershey and E.N. ess, Analyss of capacty decsons for progressve patent care hosptal facltes, Health ervces Research 5 (980), []. E. El-Darz, C. Vaslaks, T. Chaussalet and P.H. Mllard, A smulaton modellng approach to evaluatng length of stay, occupancy, emptness and bed blockng n a hosptal geratrc department, Health Care Management cence (998),

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