Finding Optimal System Time for Message Processing in Free-Sequence Multi-Stage News Agency

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1 Recent Advances n Mathematcal Methods and Computatonal Technques n Modern Scence Fndng Optmal System Tme for Message Processng n Free-Sequence Mult-Stage News Agency ABDULLAH ABDUL ABBAR *, NASHAT FORS, SHERIF RABIA Department of Producton Engneerng - Industral Engneerng, Alexandra Unversty, Alexandra EGYPT Abdullah_aj_@hotmal.com, nashatfors@gmal.com, shrfraba@hotmal.com, SERA ABED Department of Industral Engneerng, Faculty of Engneerng ng Abdul Azz Unversty, eddah ngdom of Saud Araba Abed.seraj@hotmal.com Abstract - Usually the workflow of news makng conssts of many stages ncludng Interpretng. Each stage conssts of many stages and can be dvded nto specfc processng procedure. Through experts Interpretng new perspectves and nsghts can be contrbuted to the dscourse, openng more space for understandng. The proposed study descrbes a novel multcrtera optmzaton framework for dervng optmal number of processors n the Interpretng dvson to acheve effcent, tmesavng and low-cost operaton of the system. The task of nterpretng dvson s formulated as an optmzaton problem of sequencng jobs n a freesequence, consderng the total set up cost and total tardness penalty per hour as the crtera for optmzaton. In vew of ts (INLP) nature, Computatonal experments could not assure optmum solutons, nstead, effcent solutons of the objectves are presented. eywords: mathematcal modelng, mult-stage, news agency, queung theory.. Introducton The majorty of the news agences employ human processors to process data as to perform techncal operatons; as wrtng, abstractng, edtng or newsletter desgn and dssemnaton. Humans are thought as hghly flexble processors wth stochastc servce tmes. Such systems possess a dversty of performance measures as processor utlzaton, average system tme, average watng tme, etc. Optmzaton would target fndng dynamc rules and system confguratons (or reconfguratons) to optmze the sought performance measures under stochastc nput and servce tmes. In nterpretng dvson of a news agency, the processors receve news messages from ther correspondents all over the world by tunng up ther recevers to certan channels, one rado channel for each recever. All messages are processed through four dfferent servce stages connected n seres n the same sequence, each stage performs certan operatons and s dedcated to a specfc task (Lstenng, Identfcaton, Interpretng, and Reportng), and whereby each stage comprses of parallel processors and the servce rates are dfferent from one stage to another. Each message has to pass through all the stages to be n ts fnal form. The possblty of modelng ths dvson for effcent total system tme was nvestgated by predctng watng tme n queue before startng servce n and fnally fnds the effcent number of processors at each stage n such a way that messages are not crumblng before the employees. Logcally by ncreasng the number of servers, the total system tme wll be reduced, but on the other hand the total costs of employees wll be rased. The novelty of the present research comes from the unqueness of applcaton n meda area. It could be consdered as Mult-Stage system. All stages are connected together to *Correspondng author. Tel.: ; Fax: E-mal address: Abdullah_aj_@hotmal.com ISBN:

2 Recent Advances n Mathematcal Methods and Computatonal Technques n Modern Scence predct the fnal measures, whereas many servce operatons are n a seres of stages, but researchers n the servce ndustry usually assumed that there s no relatonshp between stages[6]. Unfortunately few of the researchers based ther studes on the servce ndustry and used queung network theory. Though Zhao Xaohu et.al. [8] who proposed a new workflow smulaton method n the servce ndustry and Luca Cavallaro and Mraz [] who descrbed an approach for the performance predcton of servce based applcatons, they dd not touch upon the crtera of ths study. Specfcally Gue evn R. and m Hyun Ho [2] and Whtt Ward [7] studed a very close system to the system of ths paper, that s, call centers - as example of servce systems- n dfferent ways and both studes were to fnd the watng tme of a customer n sngle stage and mult-stage systems whch s a part of our crtera. Lao Shuangqng et. al. [3] studed non-statonary arrval rate n a mult-perod sngle-shft call center staffng problem, and used very close crtera to the man crtera of ths paper, but they used dfferent programmng approach. Abdul abbar et al. [9] studed the same system but used a fxed-sequence for all the messages from one recever. The man objectve was to mnmze setup cost, and tardness penalty per hour. Abdul abbar et al. [0] studed the same system to mnmze setup cost and tardness penalty per hour, but based only on average processng tme of a lne n a message. Abed and Abdul abbar [] studed the same system wth the same objectve but used the concept of varablty wthout usng replcatons. Abed and Abdul abbar [2] studed the same system wth the same objectve and added the concept of replcatons whch s another concept of smulaton. The present mathematcal models consder four stages of servce network, followng the dea of consderng N processors (havng ndvdual servers or clusters of servers) as presented n Mchael [4]. The arrval process of messages and processng rate of processors are modeled by a statonary and a non-statonary stochastc process. The total system tme and number of processors are the system measures. Each stage conssts of dentcal parallel processors. Each message moves through the four stages wth dentcal sequence. In each stage only one message can be processed by a processor at a tme. Meanwhle, each message can only be processed by one processor at each stage. Ths scenaro, allocates messages from any recever to any processor n each stage, assumng a free-sequence for all the messages from all recevers. The objectve s to mnmze setup cost, and tardness penalty per hour. In the begnnng the basc mathematcal model has been developed and then four dfferent Heurstc algorthms are used to solve the problem wth dfferent assumptons. In all the Heurstc algorthms, the objectve functon and constrants are not changeable, but the relatons are modfed due to the assumptons of every algorthm. Every Heurstc algorthm resulted n dfferent objectve functon value and gave dfferent effcent solutons. 2. Methodology Fve heurstc algorthm are developed to solve the problem of mult-stage mult-processor servce system n dfferent concepts. The frst Heurstc algorthm uses average arrval rates and processng rates to calculate the objectve functon and consdered to be the basc mathematcal model and ths represents steady arrval rates and steady processors modes. The second Heurstc algorthm uses random processng tmes whch presents the processors wth non-statonary modes or sklls. The thrd Heurstc algorthm uses random arrval rate whch represents non-statonary arrval rates. The fourth Heurstc algorthm uses random arrval rates and random processng rates and that represents the processors wth non-statonary modes or sklls and non-statonary arrval rates. The ffth Heurstc algorthm uses random arrval rates and random processng rates and uses teratons to represent more processng hours. Algorthm : Solvng the Basc Mathematcal Model by Averagng the Arrval Rate and Processng Rate A multstage servce system (news agency) s defned by the set ϵ {,..,4} processng stages. Each stage s a set composed of j ϵ {,,} dentcal processors. Stage composed of k ϵ {,,} recevers. A stochastc number of messages have to be processed at the four stages and one of j dentcal processor at each stage. Each message has a processng tme, PT jk, PT jk 0. The th stage can be commenced only after the completon of (-) prevous stage from the sequence. Each processor can process only one message at a tme. All the message have a free path, that s every message can processed by any processor n stages 2,3,4, but n stage all the messages from one recever should be processed ISBN:

3 Recent Advances n Mathematcal Methods and Computatonal Technques n Modern Scence by a specfc processor only. Thus, the basc mathematcal model s defned as follows: Decson varables and parameters = ndex of stages =,2,.,4 j = ndex of processors j =,2,., k = ndex of recevers k =,2,.., t = ndex of messages t=,2,.., NP Number of processors n stage NP Total number of processors n all the stages NR Number of recevers TST Total system tme TRST Total real system tme SC A gven value representng the setup cost of one processor n stage TSC A gven value representng the total setup cost TP A gven value representng tardness penalty of one watng tme unt TTP Total Tardness penalty TPPH Total tardness penalty per hour WAIT Watng tme n stage WAIT Watng tme n stage other than stage TWT Total Watng tme PT jk Processng tme for messages of recever k at processor j n stage PT jt Processng tme for message t at processor j n stage except stage TPT Total processng tme of a message U j Utlzaton of processor j n stage. APU Average processor utlzaton n stage ASU Average system utlzaton. λ k Gven Arrval rates of recever k n stage Total number of messages arrve to stage Gven total number of messages arrve to stage Pλ j Pooled arrval rates to processor j n stage µ j gven average servce rates of a processor j n stage M A large value X j = y jk = f processor j n stage s used 0 otherwse f recever k s assgned to processor j n stage 0 otherwse Relatons: = = λk () Pλ jk j =,2,., =,2,..,4 (2) TST = TPT + TWT (3) WAIT = (( Pλ jk ) ( µ Pλ jk ))( y jk λ jk ) / j= k =,2,..., (4) j j WAIT = TWT = WAIT + NP n / ( / n!)( + NP n= (. µ ( / NP 4 j ( / NP!)( ( NP )! + / I= 2 = (( NP)( µ 2 ) (( NP)( µ ) ( 4 = 2 PT jk = PT = ( ) (5) WAIT (6). y.λ jk j jk jk j= k =,2,..., (7) PT jt = PT jt = ( j ) j=. y t =,2,...,T =2,.,4 (8) TPT = ( PTk / ) + ( PTjt / ) t= =,2,.., 4, j =,2,..., (9) U j = Pλ j j j =,2,., =,2,.., 4 (0) APU = ( Uj / NP ) j= =,2,.., 4 () 4 ASU = ( U j / NP) NP = NP TSC (2) = j= X j j= jt,=,2,.., 4 (3) 4 = NP (4) = 4 = ( SC j. NP ) = j=,.., (5) TTPH = TTP. 4 (6) The Objectve Functon: Mn TSC + TTPH (7) S.T.: Pλ j = ( λjky jk ) j =,2,., (8) Pλ j µ j, j =,2,., (9) Pλ j = t= y jt j =,2,.,, = 2,..., 4,t=,, (20) µ j j =,2,.,, = 2,..., 4 (2) Pλ j ISBN:

4 Recent Advances n Mathematcal Methods and Computatonal Technques n Modern Scence y jk = j=,k =,2,., (22) y jt j= = t =,2,.,T =2,.., 4 (23) y jk M. X j, j =,2,., (24) R r= y jr M. X j j =,2,., =2,.., 4 (25) TTP = TP. ( TST TRST ), (TST TRST) > 0 0, otherwse (26) y jk = 0, k =,2,..., j=,2,.., (27) y jt = 0, t =,2,...,T j=,2,.., =2,..,I (28) X j = 0, j=,2,.., =,2,..,I (29) (NP, PT k, U j ) 0 (30) k =,2,..., j=,2,..,=,2,..,i,t =,..,T Relaton () calculates the sum of the arrval rates of all the recevers n stage. Relaton (2) calculates the sum of the arrval rates of all the recevers n every stage. Relaton (3) ensures that the average system tme s equal to the sum of the processng tmes plus the watng tmes n all the stages. Relaton (4) calculates the total watng tme at all the processors n stage (usng the watng tme formula for M/M/ queue: w q = λ/µ(µλ) [5]). Relaton (5) calculates the total watng tme at all the processors n stages 2, 3, 4 (usng the watng tme formula for M/M/S queue []). Relaton (6) calculates the total watng tme at all the processors n all the stages. Relaton (7) calculates the total processng tme of a message from recever k n stage (havng t s =/µ). Relaton (8) calculates the total processng tme of messages r n stage 2, 3, 4 (usng the processng tme formula from the queung theory). Relaton (9) calculates the Average processng tme of a message. Relaton (0) calculates the processor's utlzaton n each stage. Relaton () calculates the Average processor's utlzaton n each stage. Relaton (2) calculates the Average system utlzaton. Relaton (3) calculates number of processors n each stage. Relaton (4) shows the total number of processors. Relaton (5) calculates the total set up cost of assgnng processors to messages. Relaton (6) calculates the total tardness penalty per hour. The objectve functon (7) s to mnmze the total setup costs of usng processors and the total tardness penalty per hour. Constrant (8) ensures that the total arrval rate of a processor j n stage s equal to the sum of the arrval rates of the assgned recevers. Constrant (9) shows that the total arrval rate of a processor j n stage should be less than or equal to ts servce rate. Constrant (20) ensures that the total arrval rate of a processor j n stage 2, 3, 4 s equal to the sum of the assgned messages. Constrant (2) Show that the total number of arrvals assgned to all the processors should not exceed the total number of arrvals. Constrant (22) shows that every message from a recever k n stage should be assgned to one processor j only. Constrant (23) shows that every message t n stage 2, 3, 4 should be assgned to one processor j only. Constrant (24) fnds out whether a processor j s used n stage. Constrant (25) fnds out whether a processor j s used n stage 2,3,4. Constrant (26) calculates the total tardness penalty (f the optmzed system tme s greater than the real world system tme, there wll be a penalty and zero otherwse). and Constrant (27), (28), (29) and (30) represent the state of the varables. Algorthm -2 : Consderng Random Processng Rate Ths Heurstc algorthm dffers from algorthm- by usng random processng rates. A formula uses generated pseudo random numbers was allocated to calculate random processng rates wthn a range of ± 25% of the average processng rate. The mathematcal model would be modfed due to the new assumpton. Algorthm -3 : Consderng Random Arrval Rate Ths Heurstc algorthm dffers from algorthm- by usng random arrval rates. Another formula uses generated pseudo random numbers was allocated to calculate random processng rates wthn a range of ± 25% of the average arrval rate because the data effectuated ths range. The mathematcal model would be modfed due to the new assumpton. Algorthm -4 : Consderng Both Random Processng Rate and Random Arrval Rate Ths Heurstc algorthm dffers from algorthm- by combnng algorthm-2 and algorthm-3 by ISBN:

5 Recent Advances n Mathematcal Methods and Computatonal Technques n Modern Scence usng random processng rate plus random arrval rate. The mathematcal model would be modfed due to the new assumpton. Algorthm-5 : Consderng Both Random Processng Rate and Random Arrval Rate Wth Iteratons Ths Heurstc algorthm dffers from the algorthm- by followng the same procedure of algorthm-4 and makng several teratons. The mathematcal model would be modfed due to the new assumpton. 3. Results A. Numercal Experment To test the effcency of the proposed algorthms, all the mathematcal models were wrtten n LINGO 3 software, and numercal examples were used to solve the problem. By usng a PC wth Intel (R) Xeon (R) CPU 3.33 GHz (2 Processors) and 48.0 GB RAM, the runtmes for the fve algorthms were 00:00:57, 00:0:3, 00:5:27, 00:04:8,and 00:26:38 hours respectvely. Table 3. shows the local optma of the four algorthms usng the followng smplfed example: Number of recevers = 6 Arrval rate of Recever = 4 news transcrpts/hour Arrval rate of Recever 2 = 8 news transcrpts/hour Arrval rate of Recever 3 = 6 messages/hour Arrval rate of Recever 4 = 4 news transcrpts/hour Arrval rate of Recever 5 = 3 news transcrpts/hour Arrval rate of Recever 6 = 2 news transcrpts/hour Number of processors n stage = 6 Number of processors n stages 2,3, and 4 = 4 each Total real system tme = 0.43 hour Number of Iteratons = 2 (for logarthms wth teratons) SC λ = total arrval rate (message/hour) µ = processng rate (message/hour) NP = number of processors SC = setup cost of one processor 4. Dscusson By comparng the results of the optmzaton for the fve algorthms, t s found that the mnmum objectve functon at SR 480 s acheved by algorthm 2 n whch the savng s 50%, Total system tme of about hour, a reducton of 50% n number of processors, and 70% average processor's utlzaton. Detals of comparson are presented n the followng fgures. Detals of comparson are presented n the followng fgures (Objectve Functon (SR Algorthm Number of Processors Algorthm B. Solutons Local optma are resulted from the optmzaton and not global optma because ths study s dealng wth nteger non lnear programmng (INLP), and ths can be consdered as an effcent soluton. Table : system parameters of every stage Parameters Stage Stage2 Stage3 Stage4 λ µ NP ,6 0,4 0,2 0 (Total System Tme (hour Algorthm ISBN:

6 Recent Advances n Mathematcal Methods and Computatonal Technques n Modern Scence %75 %70 %65 %60 (Savng (SR Algorthm System Utlzaton Algorthm 5. Conclusons In ths paper, a group of heurstc approaches for Multstage Servce Systems (Interpretng Dvson n A News Agency) wth Stochastc nput s proposed. Fve heurstc algorthms have been proposed. Every heurstc has a dfferent assumpton. Mathematcal models are appled for every heurstc, a computatonal example s presented n ths paper, a soluton approach for nteger nonlnear programmng problems to reduce tardness penalty and setup cost s presented too. The objectve of the study was acheved by the proposed models, number of processors, set up costs and tardness penalty are reduced by notceable percentages, and the entre models have been verfed and valdated. References [] Cavallaro Luca and Matteo Mraz. (2007). Performance evaluaton of Web Servces. phd course Advanced topcs n computer system performance analyss, poltecnco d Mlano, Department of electronc and nformaton, Italy. [2] Gue evn, and m Hyun Ho. (20). Predctng Departure Tmes n Mult-Stage Queueng Systems. Computers & Operatons Research, 39 (202), [3] Lao Shuangqng. (20). Staffng a Call Center wth Uncertan Non-Statonary Arrval Rate and Flexblty. OR Spectrum, 0.007/s [4] Mantz Mchael. Queung-model based analyss of assembly lnes wth fnte buffers and general servce tmes. Computers and Operatons Research, 35(8), [5] Nan Phlppe. (998). Basc Elements of Queung Theory: Applcaton to the Modellng of Computer Systems. Unversty of Massachusetts, U.S.A. [6] Tsung Fugee, Yantng L and Mng n. (200). Statstcal process control for multstage manufacturng and servce operatons: a revew and some extensons. Internatonal ournal of Servces Operatons and Informatcs. [7] Whtt Ward. (999). Predctng Queueng Delays. Management Scence, 45(6), [8] Zhao Xaohu, Dehen Zhan, and Xaofe Xu. (2006). Smulaton Method of Multple Workflows. Department of Computer Scence, Harbn Insttute of Technology, P.R.Chna. [9] Abdullah A. Abdulabbar, Nashat A. Fors, Seraj Y. Abed, and Sherf I. Raba. (203). Optmzng Message Total System Tme n a Fxed-Sequence Multstage News Agency Usng Mathematcal Modelng. Department of Producton Engneerng, Industral Engneerng, Alexandra Unversty, Alexandra, Egypt. [0] Abdullah A. Abdulabbar, Nashat A. Fors, Seraj Y. Abed, and Sherf I. Raba. (203). Predctng Optmal System Tme Based on Number of Lnes n A Message n Free-Sequence Mult-Stage News Agency. Department of Producton Engneerng, Industral Engneerng, Alexandra Unversty, Alexandra, Egypt. [] Abed Sraj and Abduljabbar Abdullah. (203). Implementng A Smulaton Concept n Mathematcal Modelng When Predctng Optmal Total System Tme. Department of Producton Engneerng, Industral Engneerng, Alexandra Unversty, Alexandra, Egypt. [2] Abed Sraj and Abduljabbar Abdullah. (203). Implementng The Concepts of Varaton of The Input Varables and Replcatons Used n Smulaton n Mathematcal Modelng. Department of Producton Engneerng, Industral Engneerng, Alexandra Unversty, Alexandra, Egypt. ISBN: