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A multi-server queueing-inventory system with stock-dependent demand
9th IFAC Conference on Manufacturing Modelling, Management and 9th IFAC Conference on Manufacturing Modelling, Management and Control 9th IFAC Conference on Manufacturing Modelling, Management and Control Available online at www.sciencedirect.com 9th IFAC Conference on Manufacturing Modelling, Management and Berlin, Germany, August 28-30, 2019 Control 9th IFAC Conference on Manufacturing Modelling, Management and Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019 Berlin, Germany, August 28-30, 2019
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IFAC PapersOnLine 52-13 (2019) 671–676
A multi-server queueing-inventory system with stock-dependent demand A multi-server queueing-inventory system with stock-dependent demand A multi-server queueing-inventory system with stock-dependent demand A multi-server queueing-inventory system with stock-dependent demand 1 2 system with stock-dependent 3 4 A multi-server queueing-inventory demand
Gabi Gabi Hanukov Hanukov11,, Tal Tal Avinadav Avinadav22,, Tatyana Tatyana Chernonog Chernonog33,, Uri Uri Yechiali Yechiali44 Gabi Hanukov11, Tal Avinadav22, Tatyana Chernonog33, Uri Yechiali44 Avinadav2, Tatyana Chernonog3, Uri Yechiali4 Gabi Hanukov 1 1, TalUniversity, Bar-Ilan Gan,Chernonog 5290002, Israel, [email protected] 11Department of Management, Gabi Hanukov , TalUniversity, Avinadav Ramat , Tatyana , Uri Yechiali of Management, Bar-Ilan Ramat Gan, 5290002, Israel, [email protected] 1Department 2 of of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] 22Department 1Department Department of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] Department of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] 2 3 1Department ofManagement, Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] Department of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] of Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] 33Department 2 Department of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] Department of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] 2 3 4 Department ofand Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] of Research, School of Sciences, Tel Aviv 4 3Department 4Department of Statistics Statistics and Operations Operations Research, SchoolRamat of Mathematical Mathematical Sciences, [email protected] Aviv University, University, Tel Tel Aviv, Aviv, 6997801, 6997801, of Management, Bar-Ilan University, Gan, 5290002, Israel, 3Department 4Department Department of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel, [email protected] 4 Israel, [email protected] School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 6997801, 4Department of Statistics and Operations Research, Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel, [email protected] Israel, [email protected] Israel, [email protected] Abstract: Abstract: We We consider consider aa two-server two-server service service system system in in which which idle idle servers servers produce produce and and store store preliminary preliminary Abstract: We consider a sojourn two-server service system customers in which idle servers produce and store preliminary services for reducing the time of incoming and increasing customers arrival rate. services forWe reducing thea sojourn timeservice of incoming and servers increasing customers arrival rate. We We Abstract: consider two-server system customers in which idle produce and store preliminary Abstract: We consider a sojourn two-server service system inawhich idle servers produce and store preliminary services forsystem reducing the time of incoming customers andbased increasing customers arrival rate. We model the as a Markovian process and provide method, on matrix geometric (MG) analymodel theforsystem as athe Markovian process and provide a method, on matrix geometric (MG) analyservices reducing sojourn time of incoming customers andbased increasing customers arrival rate. We services forsystem reducing sojourn time of incoming customers andbased increasing customers arrival rate. We model the as athe Markovian process and provide aprobabilities method, onrelevant matrix geometric (MG) analysis, to obtain closed-form solutions to the steady state and performance measures. sis, to the obtain closed-form solutions to the and steady state aprobabilities andonrelevant measures. model system as a Markovian process provide method, based matrix performance geometric (MG) analymodel the system as aofMarkovian process and provide aof method, based onrelevant matrix geometric (MG) analysis, to obtain closed-form solutions to the steady state probabilities and performance measures. We show the relation the elements in the rate matrix the MG to Catalan numbers, and prove that We to show the relation of thesolutions elementstointhe thesteady rate matrix the MG to and Catalan numbers, and prove that the the sis, obtain closed-form state of probabilities relevant performance measures. sis, obtain closed-form toin the state of probabilities relevant performance measures. We to show the relation of thesolutions elements thesteady rate matrix the MG to and Catalan numbers, and prove that the stability condition remains as is in the traditional M/M/2 queue, although the expected sojourn time of stability remains is in the in traditional M/M/2of queue, although the expected timethat of cuscusWe showcondition the relation of theaselements the rate matrix the MG to Catalan numbers,sojourn and prove the We show the relation of the elements in the rate matrix of the MG to Catalan numbers, and prove that the stability condition remains as is in the traditional M/M/2 queue, although the expected sojourn time of customers been provide an analysis for aa system in the PS tomers has has been reduced. reduced. We provide an economic economic analysis for although system the in which which the sojourn PS capacity capacity and the stability condition remainsWe as is in the traditional M/M/2 queue, expected time and of cusstabilityhas condition remains as is inarrival the traditional M/M/2 queue, the expected time and of customers been reduced. We provide an rate economic analysis for although a system in which the sojourn PS capacity the investment to increase customers' are decision variables. Copyright © 2019 IFAC investment to increase customers' arrival are decision variables. Copyright © 2019 IFAC tomers has been reduced. We provide an rate economic analysis for a system in which the PS capacity and the tomers has been reduced. We provide an rate economic analysis for a system in which the PS capacity and the investment to increase customers' arrival are decision variables. Copyright © 2019 IFAC investment to increase customers' arrival are system, decision variables. ©Ltd. 2019 © 2019, IFAC (International Federation of rate Automatic Control) Hosting Copyright by Elsevier AllIFAC rights reserved. Keywords: Markov process, queueing-inventory stock-dependent demand, matrix geometric investment increase customers' arrival rate are system, decision variables. Copyright © 2019 IFAC Keywords: to Markov process, queueing-inventory stock-dependent demand, matrix geometric analanalKeywords: Markov process, queueing-inventory system, stock-dependent demand, matrix geometric analysis, economic analysis ysis, economic analysis Markov Keywords: process, queueing-inventory system, stock-dependent demand, matrix geometric analKeywords: Markov process, queueing-inventory system, stock-dependent demand, matrix geometric analysis, economic analysis ysis, economic analysis Urban ysis, economic analysis Urban 2005) 2005) that that customers customers buy buy more more units units when when the the level level of of Urban 2005) that customers buy more units when the level of 1. INTRODUCTION the presented inventory is higher due to marketing effects 1. INTRODUCTION the presented inventory is higher dueunits to marketing effects buy more when the level of Urban 2005) that customers Urban 2005) that customers buy more when the level of 1. INTRODUCTION the presented inventory is higher dueunits to marketing effects (e.g., a wider selection). In our model, the stock of prelimi(e.g.,presented a wider selection). In higher our model, stock of prelimithe inventory is due the to marketing effects 1. INTRODUCTION Reducing customers1.waiting times in service systems is a INTRODUCTION the presented inventory is higher due to marketing effects (e.g., a wider selection). In our model, the stock of preliminary services has an additional positive effect on customers Reducing customers waiting times in service systems is a (e.g., nary services an additional effect onofcustomers a wider has selection). In our positive model, the stock prelimiReducing objective, customers which waitingaims times in service systems is a (e.g., common to reduce costs associated a wider selection). our positive model, the stock ofcustomers preliminary services has an additional effect onthe demand since it implies aaInshorter sojourn time in system. common to in reduce costs associated Reducing objective, customers which waitingaims times service systems is a nary demand since it implies shorter sojourn time in the system. services has an additional positive effect on customers Reducing customers waiting times in service systems is a demand common objective, which aims tosystem reduce costs associated with customers sojourn time in the and increase their nary services has an additional positive effect on customers since it implies a shorter sojourn time in the system. To model this effect, we use state-dependent customers' arriwith customers sojourn timeaims in thetosystem increase their demand common objective, which reduceand costs associated To model this effect, we use state-dependent customers' arrisince it implies a shorter sojourn time in the system. common objective, which aims tosystem reduce costs associated with customers sojourn time in the and increase their satisfaction. Usually, it is achieved by using faster serves demand since it implies a shorter sojourn time in the system. To model this effect, we use state-dependent customers' arrival rate (Gupta 1967, Winston 1978, Guo and Hassin 2011, satisfaction. Usually, is achieved by using faster serves with customers sojournit time in the system and increase their To val rate (Gupta 1967, Winston 1978, Guo and Hassin 2011, model this effect, we use state-dependent customers' arriwith customers sojourn inand the Zhang system and increase their To satisfaction. Usually, it time isGuo achieved by using faster serves (e.g., Hwang et al. 2010, 2013) or by hiring model this effect, we use state-dependent customers' arrival rate (Gupta 1967, Winston 1978, Guo and Hassin 2011, Zhao and Lian 2011). (e.g., Hwang Usually, et al. 2010, and Zhang 2013)faster or by serves hiring val satisfaction. it isGuo achieved by using Zhaorate and(Gupta Lian 2011). 1967, Winston 1978, Guo and Hassin 2011, satisfaction. Usually, it isGuo achieved by using faster (e.g., Hwang et al. 2010, and Zhang 2013) orcostly. by serves hiring additional servers. However, these methods are Reval rate (Gupta 1967, Winston 1978, Guo and Hassin 2011, Zhao and Lian 2011). additional servers. methods are orcostly. Re- Zhao and Lian 2011). (e.g., Hwang et al. 2010, Guo these and Zhang 2013) by hiring However, (e.g., Hwang et al. 2010, Guo these and2018a,b) Zhang 2013) orcostly. by hiring additional servers. However, methods aresuggested Recent (Hanukov et have to Our is and Lian 2011). motivated cent works works servers. (Hanukov et al. al. 2017, 2017, 2018a,b) have to Zhao Our model model is mostly mostly motivated by by the the fast fast food food industry industry in in additional However, these methods aresuggested costly. Readditional servers. However, these methods are costly. Recent works (Hanukov et al. 2017, 2018a,b) have suggested to Our model issuch mostly motivated patties by the or fast foodpizza, industry in use a server's idle time to produce and stock preliminary serwhich food, as hamburger basic can use aworks server's which food,issuch as hamburger basic can be be cent (Hanukov ettoal.produce 2017, 2018a,b) have suggestedserto Our model mostly motivated patties by the or fast foodpizza, industry in idle time and stock preliminary cent (Hanukov et 2017, 2018a,b) suggested to Our model issuch mostly motivated by the fast foodthe industry in as hamburger patties or basic pizza, can be use aworks server's idle time toal.produce and stockhave preliminary serwhich food, vices, which will be used to serve faster future customers, prepared before demand occurs, and only upon arrival of vices, which will be used to serveand faster customers, prepared before occurs, patties and only upon the arrival of use a server's idle time to produce stockfuture preliminary ser- which food, suchdemand as hamburger or basic pizza, can be use athus server's idle the time to produce and stock preliminary ser- prepared which food, suchdemand as heated hamburger patties oraccording basic pizza, can be before occurs, and only upon the arrival of vices, which will be used to serve faster future customers, and increase productivity of the service system witha customer they are up, adjusted to specific and thus increase of the service with- aprepared customer they demand are heated up, adjusted specific vices, which will the be productivity used to serve faster futuresystem customers, before occurs, and onlyaccording upon thetoarrival of customer they demand are heated up, adjusted according specific vices, which will be productivity used to serve faster futuresystem customers, before occurs, andtoonly upon thetoarrival of and thus increase the of the service with- aprepared out in or resources. innovaand served customer. out investing investing in additional additional or better better resources. This innovarequirement, andup, served to the the customer. Another and thus increase the productivity of the service This system with- customers acustomers customerrequirement, they are heated adjusted according to Another specific and investing thus increase the productivity of the service This system with- customers a customer they are heated adjusted according toretailing; specific requirement, andup, served to the customer. Another out in additional or better resources. innovative approach is modelled using a queueing-inventory system domain in which our model is applicable is bicycle tive investing approach in is modelled a queueing-inventory system customers domain in requirement, which our model applicable bicycle retailing; out additionalusing or better resources. This innovaand is served to the iscustomer. Another out investing additional or better resources. This(see innovacustomers and is served to the is Another domain in requirement, which our model applicable bicycle retailing; tive approachbyin is modelled using a queueing-inventory system represented two-dimensional Markov e.g. can assemble parts aacustomer. before represented two-dimensional Markov process process (see e.g. where where mechanics mechanics canmodel assemble parts of of is before aa tive approachbyisaamodelled using a queueing-inventory system domain in which our is applicable bicycle retailing; tive approach using a queueing-inventory system domainmechanics in which our model is applicable retailing; can assemble partsassemble of is a bicycle before a represented byisamodelled two-dimensional Markov process (see e.g. where Cox 2017), which is solved using a matrix geometric analycustomer's arrival, and subsequently the remaining Cox 2017), which is solved using aMarkov matrix process geometric customer's arrival, can and assemble subsequently the remaining represented by a two-dimensional (see e.g. where mechanics partsassemble of a bicycle before a analyrepresented byquestion a two-dimensional process e.g. customer's whereinmechanics assemble partsassemble of a bicycle before a Cox 2017), which is solved using aMarkov matrix geometric analyarrival, can and subsequently the remaining sis. The main in these models is how many (see prelimiwith the specific requirements parts in accordance accordance withsubsequently the customer’s customer’s specificthe requirements CoxThe 2017), which is solved using a matrix geometric analy- parts customer's arrival, and assemble remaining sis. main question in these models is how many prelimiparts in accordance with the customer’s specific requirements Cox 2017), which is solved using a matrix geometric analycustomer's arrival, and subsequently assemble the remaining sis. The main question in these models is how many preliminary services should be produced and stocked when the serv- and Handmade for are and preferences. preferences. Handmade nameplates for doors doorsrequirements are another another sis. The main should question these models is how when manythe prelimiin accordance with the nameplates customer’s specific nary services bein produced and stocked serv- parts sis.isThe main question these models iscontrast how when many prelimipartspreferences. in accordance with the customer’s specific requirements Handmade nameplates for doors arecan another nary services should beinproduced and stocked the serv- and er idle. Actually, this approach is in to extensive example in which service can be split up. The server proer is idle. Actually, this approach is in contrast to extensive example in which service can be split up. The server pronary services should be produced and stocked when the serv- and preferences. Handmade nameplates for doors arecan another nary shouldto bewhich produced and stocked when the servand preferences. Handmade doors arecan another example in which service cannameplates be splitceramic up.for The server proer is services idle.according Actually, this approach is in contrast to extensive literature servers' idle times may be used duce basic nameplates from wood, or glass (which literature according to which servers' idle times may be used duce basic nameplates from wood, ceramic or glass (which er is idle. Actually, this approach is in contrast to extensive example in which service can be split up. The server can proer is idle.according Actually, thiswhich approach is in contrast to extensive example in which service can wood, be splitceramic up. Theorserver probasic nameplates from glasscan (which literature to servers' idle times may be used duce for out (also as identical) before an made, the for carrying carrying out ancillary ancillary duties (also known as 'vacation 'vacation are not not identical) beforefrom an order order is ceramic made, and and complete the literature according to whichduties servers' idleknown times may be used are duce basic nameplates wood,is or complete glass (which literature according to which servers' idle timesRosenberg may be used ducenot basic nameplates wood, or complete glass (which are identical) beforefrom an order is ceramic made, and the for carrying out ancillary duties (also known as 'vacation models' in e.g. Levy and Yechiali 1975, 1976; and nameplate upon request (e.g., writing the name, add decoramodels' Rosenberg and are nameplate upon request writing the name, add decorafor carrying out ancillary duties 1975, (also 1976; known as 'vacation not identical) before (e.g., an order is made, and complete the in e.g. Levy and Yechiali for carrying ancillary (also known as Yang 'vacation nameplate are not identical)request before (e.g., an order is made, and complete the writing the name, add decoramodels' e.g.out Levy andetYechiali 1976; Rosenberg Yechiali Boxma al.duties 2002;1975, Yechiali 2004; Yechialiin 1993; Boxma Yang and tion, etc.). etc.). upon models' in1993; e.g. Levy andetYechiali 1976;2004; Rosenberg and tion, nameplate upon request (e.g., writing the name, add decoraal. 2002;1975, Yechiali etc.). upon request (e.g., writing the name, add decoramodels' in1993; e.g. Levy andet Yechiali 1975, 1976; Rosenberg and tion, nameplate Yechiali Boxma al. 2002; Yechiali 2004; Yang Wu 2015; Mytalas and Zazanis 2015; and Guha et al. 2016) Wu 2015;1993; Mytalas and et Zazanis 2015; and Guha al. 2016) Yechiali Boxma al. 2002; Yechiali 2004;et Yang and tion, etc.). 2. MODEL FORMULATION FORMULATION Yechiali 1993; Boxma al. 2002; Yechiali 2004;et Yang and tion, etc.). Wu 2015; Mytalas and et Zazanis 2015; andtimes. Guha al. 2016) which may increase waiting far, this 2. MODEL which mayMytalas increaseandcustomers customers waiting So far,2016) this Wu 2015; Zazanis 2015; andtimes. Guha So et al. 2. MODEL FORMULATION Assume a service system with two two servers, servers, in in which which fullfullWu 2015; Mytalas and Zazanis 2015; and Guha et al. 2016) which may increase customers waiting times. So far, this innovative approach has been applied (analytically) a service system FORMULATION with 2. MODEL innovative (analytically) only for which mayapproach increasehas customers waiting times. Soonly far, for thisaa Assume been applied 2. is MODEL FORMULATION Assume a service system with two servers, with in which fullwhich mayapproach increase customers waiting times.Poisson Soonly far,(custhisa service innovative has been applied (analytically) for service (FS) time exponentially-distributed mean 1/ single-server service systems and homogenous a service with two servers, with in which ,, (FS) time issystem exponentially-distributed mean full1/ single-server service has systems and homogenous Poisson innovative approach been applied (analytically) only (cusfor a Assume Assume a service witha Poisson two servers, in with which full(FS) time issystem exponentially-distributed with mean 1/ , innovative approach has been applied (analytically) only (cusfor a service single-server service systems and homogenous Poisson and customers arrival follows process rate tomer) arrival process. service (FS) time is exponentially-distributed withwith mean ,.. customers arrival follows a Poisson process rate1/ tomer) arrivalservice process.systems and homogenous Poisson (cus- and single-server service (FS) can time is split exponentially-distributed withwith mean ,. single-server and customers arrival follows a Poisson process rate1/ tomer) arrivalservice process.systems and homogenous Poisson (cus- The The service be into two phases: preliminary service service canarrival be split into two phases: process preliminary and customers follows a Poisson with service rate . tomer) arrival process. The current work applies the above innovative approach in and customers arrival follows a Poisson process with rate . tomer) arrival process. The service can be split into two phases: preliminary service (PS), which can be carried out without the presence of cusThe current work applies the above innovative approach in (PS), which can can be be split carried cusservice intoout twowithout phases:the preliminary service presence of The current work applies the above innovative approach in The the framework of a multi-server service system with stockThe service can be split into two phases: preliminary service (PS), which can be carried out without the presence of customers, and complementary service (CS), which can be given the framework The current work the above innovative in tomers, of aapplies multi-server service system approach with stockand complementary service (CS),the which can beofgiven which can be carried out without presence cusThe currentcustomers' work the above innovative in (PS), the framework of aapplies multi-server system approach with stockdependent arrival Multi-server are (PS), which be carried out without the ofgiven customers, and complementary service (CS), which can beis only when acan customer is present. present. When apresence server idle, dependent customers' arrival rate. rate.service Multi-server systems are only the framework of a multi-server service system systems with stockwhen a customer is When a server is idle, tomers, and complementary service (CS), which can be given the framework ofreal a multi-server system with stockdependent customers' arrival rate.service Multi-server systems are tomers, widely spread in life businesses, but in case of queueingand complementary service (CS), which can be given only when a customer is present. When a server is idle, he/she can produce PSs. In line with prior works in this dowidely spread in real life businesses, but in case of queueingdependent customers' arrival rate. Multi-server systems are he/she can produce PSs. is In present. line withWhen prior works in this dowhen a customer a server is idle, dependent customers' arrival rate. Multi-server systems are only widely spread in real life businesses, but in case of queueinginventory systems are considerably complicated to only when a customer is When a server is et idle, he/she can produce PSs. In present. line withet prior works in this domain (Benjaafar et al. 2011, Flapper al. 2012, Iravani al. inventory systems arelife considerably more complicated to anaana- main widely spread in real businesses,more but in case of queueinget al. 2011, Flapper al. 2012, al. he/she(Benjaafar can produce PSs. In line withetprior worksIravani in thisetdowidely spread in real businesses, butit in case of queueinginventory systems arelife considerably more complicated to analyze. As for stock-dependent demand, is known (see, e.g., he/she can produce PSs. In line with prior works in this domain (Benjaafar et 2011, Flapper et al. 2012, Iravani et al. 2012, Hanukov et al. 2017), the PS's preparation time is aslyze. As for stock-dependent demand, it is known (see, e.g., 2012, inventory systems are considerably more complicated to anaHanukov et al. 2017), the PS's preparation time is as(Benjaafar et 2011, Flapper et al. 2012, Iravani et al. inventory systems are considerably more complicated to analyze. As for stock-dependent demand, it is known (see, e.g., main main (Benjaafar et 2011, Flapper et al. 2012, Iravani et al. 2012, Hanukov et al. 2017), the PS's preparation time is aslyze. As for stock-dependent demand, it is known (see, e.g., 2012, Hanukov et al. 2017), the PS's preparation time is aslyze. As © for stock-dependent demand, it is known (see, Control) e.g., 682Hosting 2405-8963 IFAC (International Federation of Automatic by ElsevieretLtd. rightsthe reserved. 2012, Hanukov al. All 2017), PS's preparation time is asCopyright © 2019, 2019 IFAC Copyright 2019 responsibility IFAC 682Control. Peer review© of International Federation of Automatic Copyright ©under 2019 IFAC 682 10.1016/j.ifacol.2019.11.124 Copyright © 2019 IFAC 682 Copyright © 2019 IFAC 682
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Gabi Hanukov et al. / IFAC PapersOnLine 52-13 (2019) 671–676
sumed to be exponentially distributed with parameter α. The PSs are stored until the arrival of customers, and, as indicated, are used to reduce the sojourn time of future customers.
steady-state probabilities (i.e., pi, j ,k when the system is stable). Then, various performance measures, such as mean queue size, mean sojourn time, mean inventory level of PSs, mean time a PS spends in inventory, etc. are calculated. To do so, matrix geometric analysis (Neuts 1981) is applied, which entails calculation of the so-called rate matrix R (see bellow). The system’s states are arranged in the following lexicographic order (i = 2,3,…):
The total number of PSs in stock is limited to n (capacity), and when the inventoried PSs reach this capacity limit, the servers stop producing PSs and return to idle position. When a customer reaches the head of the line and a PS is available, his/her CS phase starts immediately. The CS time is exponentially distributed with mean 1/, where >. That is, the CS duration is stochastically shorter than an FS duration (first order stochastic dominance). When PSs are available, customers’ arrival rate increases to ( ) until the number of customers is equal to the number of PSs (which are both observable by the customers).
{( 0, 0, 0 ) , ( 0,1, 0 ) ,..., ( 0, n, 0 ) ; (1, 0, 0 ) , (1,1, 0 ) , (1,1,1) ,..., (1, n, 0 ) , (1, n,1) ;...;
( i, 0, 0 ) , ( i,1,1) , ( i, 2,1) , ( i, 2, 2 ) ,..., ( i, n,1) , ( i, n, 2 ) ;...}, and construct its infinitesimal generator matrix, denoted by Q: B1,1 B1,2 B2,1 B2,2 B3,1 0 0 0 Q= 0 0 0 0 0 0
The process can be formulated as a three-dimensional continuous-time Markov chain with a state space Lt , St ,Ut , where Lt 0,1, 2,..., denotes the number of customers in the system at time t, St 0,1, 2,..n denotes the total number of PSs at time t, and Ut 0,1, 2 denotes the number of customers at their CS phase at time t. Let L limt → Lt , and Let S limt → St U limt → Ut . pi, j ,k = Pr( L = i, S = j,U = k ) denote the joint probability
0
0
0
0
0
0
0
B2,3 B3,2
0
0
0
0
0
0
B3,3 B4,2
0
0
0
0
0
A2
B4,3
0
0
0
0
0
0
Bn +1,3 A1 A2
0
0 0
Bn +1,2 A2 0
0
0 0
A2 0 0
A0 A1
0 A0
, (1)
where matrices Bi, j and Ak ( k = 0,1, 2 ) are given in the Appendix.
distribution of the latter three-dimensional process. pi, j ,k
3. STEADY STATE ANALYSIS We now claim that the condition for stability of the system is identical to that of a regular M/M/2 queue without PSs (i.e., a Markovian service system with two parallel identical servers; see, e.g., Harchol-Balter 2013 p. 258): Theorem 1. The stability condition of the queuing-inventory system is 2 . Proof. The condition for stability of our system, according to Neuts (1981, p. 83) is (2) A0e A2e ,
represents the long run fraction of time that the system stays in state ( L = i, S = j,U = k ) . As an illustration, Figure 1 bellow depicts the system's states and its corresponding transition rate diagram for the case n = 3 . Each circle in the diagram depicts a state ( L = i, S = j,U = k ) and the arrows indicate the transition directions and rates.
where A0 and A2 (as well as A1 ) are given in the Appendix.
e = (1,1,...,1)T is a vector of ones, and = (1 , 2 ,..., 2n ) is the unique solution of the linear system A=0 (3) e = 1, where A A0 + A1 + A2 . In our case, = (1,0,...,0) . Thus, using A0 and A2 , the stability condition in equation (2) translates into 2 . Theorem 1 implies that although preparing PSs improves the system’s performance, it does not stringent its stability condition. This result can be further explained as follows: due to the stochastic nature of the system, at some point of time both servers will be fully occupied, and will not be able to produce any PS, so the system will become a regular M/M/2 queue. This explanation leads us to state the following conjecture:
Figure 1. System states and transition-rate diagram for n = 3.
Conjecture 1. The condition for stability of the queueinginventory system as described above having s identical servers is s .
Our goal is to obtain the system's stability condition (i.e., the condition on the parameters insuring that the queue size will not grow beyond any bound) and to derive the system’s 683
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Gabi Hanukov et al. / IFAC PapersOnLine 52-13 (2019) 671–676
0 C0.5( i − j + 3) 0.5( i − j +1) 0.5( i − j + 5) i − j +3 ( + 2 ) ( + + ) k +1 0.5(i − j −1) r i , j + 2 + 2 k C k ( 2 ) + 2k +2 ri , j = ( + 2 ) k =0 0.5( i − j −1) 0.5 ( i − j +1) − k r j + 1+ 2 k , j + 2 C 0.5 ( i − j −1) − k ( ) +2 i ( + 2 )( + + ) − j − 2 k k =0 2 0.5( i − j ) C0.5(i − j ) 0.5(i − j ) 0.5(i − j + 2) ( + 2 ) i − j +1
The matrix geometric analysis is based on derivation of a socalled rate-matrix R [ri, j ]2n2n , which is obtained by solving the quadratic matrix equation (see Neuts 1981) A0 + RA1 + R2 A2 = 02n2n .
(4)
Define the (n + 1) -dimensional vector of probabilities
(
p0 p0,0,0
p0,1,0
p0,2,0
)
p0,3,0
p0, n,0 ,
the (2n + 1) -dimensional vector of probabilities
(
p1 p1,0,0 p1,1,0 p1,1,1 p1,2,0 p1,2,1 p1,3,0 p1,3,1
p1, n,0
p1,n,1
)
and the (2n) -dimensional vectors of probabilities
(
pi pi ,0,0
pi ,1,1
pi ,2,1
pi ,2,2
pi,3,1
pi,3,2
pi, n,1
pi, n,2
i = n + 1, n + 2, n + 3,..., .
(5)
numerical calculations (see Chapter 8 in Latouche and Ramaswami 1999), mostly using successive substitutions (Neuts 1981, p. 37). However, for the current model, it is possible to derive closed-form expressions for all ri , j , as given in Theorem 2, due to the structure of the stochastic process given in Figure 1. ( 2m ) ! , Let Cm be the m-th Catalan number (i.e., Cm ( m + 1)!m! see Koshy 2008), where m is a non-negative integer. Then, Theorem 2. The entries of matrix R are: /(2 )
i =1 r22, 2 i=2 2 (1 − r2, 2 ) 0.5 ( i −1) ( + + ) ri , 2 + 2 k (r2 + 2 k , 2 + 2r2 + 2 k ,1 ) k =0 0.5(i −3) 0.5 ( i −1) − k r C ( ) 3 + 2 , 2 0 . 5 ( − 3 ) − k i k + − 3 − 2 i k k =0 i = 3,5,...,2n − 1 ( + + ) 0.5 ( i −1) − k 0.5 ( i − 5 ) r3+ 2 k ,1C0.5(i −3) − k ( ) ri ,1 = + 2 −1 ( + + ) i − 3 − 2 k k =0 2 ( + ) ( + 2 )r (2r + r ) 2 ,1 2, 2 i,2 0.5 ( i − 2 ) − k 0.5 ( i − 4 ) r C ( 2 ) + 0.5 4 + 2 k , 2 0.5( i − 4) − k i − 4 − 2 k ( + 2 ) k =0 0.5 ( i − 2 ) − k 0.5 ( i − 6 ) r4 + 2 k ,1C0.5(i − 4 ) − k (2 ) −1 i −4−2k ( 2 ) + k =0 i = 4,6,...,2n 4
ri , j
i = j − 1, j + 1,...,2n − 1 i = j , j + 2,...,2n
0 + + − ( + + ) 2 − 4 2 0.5 ( i −1) 0.5 ( i +1) 0.5 ( i − 5 ) r C ( ) 0.5(i −1) − k C0.5(i −1) + −1 3+ 2 k , 2 0.5(i −3) − k i −3− 2 k ( + + ) i − 2 + + ( ) k =0 0.5 ( i − 3) r3+ 2 k , 4 C0.5( i −3) − k ( ) 0.5(i −1) − k + 2 ( + + ) i − 3 − 2 k k =0 ri , 2 = k +1 0.5 ( i − 3) ri , 4 + 2 k ( + + )(C k (2 ) + r4 + 2 k , 2 ( + 2 ) 2 k +1 ) + ( + 2 ) 2 k +1 k =0 ( + + )(( + + ) − r2, 2 ) − 2 0.5( i − 2 ) C0.5(i − 2 ) 0.5( i − 2 ) 0.5i ( + 2 ) i − 3 0.5 ( i − 2 ) − k 0. 5 ( i − 6 ) r C ( 2 ) + k i − − k 4 2 , 2 0 . 5 ( 4 ) − 1 + (2 ) ( + 2 ) i − 4 − 2 k k =0 ( + 2 )(( + + ) − r2, 2 ) −
In most cases, the entries ri , j of the matrix R are found by
0 = 0.5( i − j ) 0.5( i − j + 2 ) C0.5( i − j ) i − j +1 ( ) + +
i = 1,2,..., j − 2
for j = 4,6,..., 2n .
)
i = 2,3, 4,... . Following Neuts (1981), the vectors pi satisfy i − n +1 pi = pn +1 R ( ) ,
673
i =1 i=2
i = 3,5,...,2n − 1
i = 4,6,...,2n
Proof. The solution has been obtained using induction, and can be verified by substitution in (5). As for calculating pi , i = 0,1,..., n + 1 , we use a subset of
equations from pQ = 0 (where p ( p0 , p1 , p2 ,...) ) combined
with
the
normalization
equa-
tion p0 e + p1 e +
p e = 1 . i
i =2
Consequently, we solve the following linear set: p0 B1,1 + p1 B2,1 = 0
p0 B1,2 + p1 B2,2 + p2 B3,1 = 0 pi −1 Bi ,3 + pi Bi +1,2 + pi +1 A2 = 0
i = 2,3,..., n
pn Bn +1,3 + pn +1 ( A1 + RA2 ) = 0 n
p0 e + p1 e +
p e + p i
n +1
I − R −1 e = 1.
i =2
i = 1, 2,..., j − 1 and i = j + 1, j + 3,..., 2n
4. PERFORMANCE MEASURES
i = j , j + 2,..., 2n − 1
Based on the steady state probabilities, we show how to calculate relevant systems performance measures. For a given capacity of n PSs, let L(n) and Lq (n) be the mean number
for j = 3,5,..., 2n −1 .
of customers in the system and in queue, respectively. Simi684
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and Tq ( n ) = Sq ( n ) / eff , respectively.
larly, let W (n) and Wq (n) be the mean sojourn time of a customer in the system and in queue, respectively; S (n) and Sq (n) be the mean number of PSs in the system and in in-
5. ECONOMIC ANALYSIS In this section, we provide an economic analysis of our queueing-inventory model. Let p be the revenue from supplying a service to a customer, so that the expected system's revenue rate is equal to peff . Let c be the sojourn cost per unit
ventory, respectively; and T (n) and Tq (n) be the mean time a PS resides in the system and in inventory, respectively.
Using
(4)
and
the
R
equations
i
−1
= I − R
and
−2
) e . (6)
of time per customer in the system, h—the holding cost per unit of time per inventoried PS (assuming no holding cost for a PS that moves on to the CS phase). Assume that the server can publicize the number of available PSs (i.e. make the stock observable) which stimulates demand due to shorter sojourn time. The publication may be generated in different levels. For example, the server can rent screen times for promotion in different locations near its store, where each screen has a potential to increase the number of potential customers. Another example is transmitting promotion messages to customers' smartphones in a larger distance (radius) to reach more customers. Thus, we denote by A( ) the cost rate of making the number of PSs observable (hereafter transparency cost) as a function of the desired arrival rate . We assume that the transparency cost is a convex increasing function of the gap in the arrival rate. Specifically, we use A( ) = a( − ) p , where a 0 and 1 . The objective is to maximize the total expected profit per time unit by controlling the PS capacity n and . Thus, the profit function is defined as follows
i =0
(i + 1) R
i
−2
= I − R , we obtain:
i =0
L ( n ) = p1e +
( n I − R
n
ip e + p
−1
n +1
i
i =2
+ I − R
Lq ( n ) = L ( n ) − 2 (1 − p0e ) + p1e .
(7)
In order to calculate the mean sojourn time of customer in system and in queue, we first have to calculate the effective customer's arrival rate eff . Let p be the proportion of time when customers arrive with rate , and p = 1 − p the proportion of time when customers arrive with rate . Then, the effective customers' arrival rate is obtained as follows (8) eff = p + p = ( − ) p + , where 1
p =
n
n
pi , j ,0 +
i = 0 j =1
n
p1, j ,1 +
j =2
n −1
n
pi , j ,1 +
i = 2 j =i
n
p
i , j ,2 .
(9)
i = 2 j =i +1
max
Then, W ( n ) = L ( n ) / eff and Wq ( n ) = Lq ( n ) / eff . In order to
obtain
S ( n) ,
we
vn +1 = ( 0 1 2 3
define
three
column
n0,1,2...
vectors
T
n n)
T
v2 n +1 = ( 0 1 1 2 2 3 3
(13)
To illustrate the behaviour of the expected profit as a function of PS capacity, n, and arrival rate when PSs are available, , we use the following parameter values: = 16 , = 10 , = 20 , = 18 , p = 0.5 , a = 0.2 , = 2 , c = 1 and h = 1 . The results are given in Table 1 and Figure 2.
n) ,
v2 n = ( 0 1 2 2 3 3
R(n, ) = peff − A( ) − cL(n) − hSq (n) .
and
n n ) . Thus, by using T
equation (4) and algebraic manipulations, we get
S ( n ) = p0 vn +1 + p1v2 n +1 +
n
p v
−1
i 2n
+ pn +1 I − R v2 n . (10)
i =2
Similarly, in order to get S q ( n ) , we define three column vectors
un +1 = ( 0 1 2 3
n) ,
n −1 n − 2)
and
u2 n = ( 0 0 1 0 2 1 3 2
T
u2 n +1 = ( 0 1 0 2 1 3 2
Sq ( n ) = p0un +1 + p1u2n +1 +
T
R(n,) 6
T
n
p u
i 2n
19
5
n n − 1) , so that
17
4
−1
+ pn +1 I − R u2 n . (11)
15
3
13 11
2
i =2
Given that inventory level is less than n, PSs are produced by the two servers when there are no customers in the system, and by one of the servers when there is only one customer in the system. Therefore, the effective production rate of PSs is eff = 2 ( p0 e − p0, n,0 ) + ( p1e − p1, n,0 − p1, n,1 ) . (12)
9
1
n
7 0
5
3
1
Figure 2. The expected profit as a function of n and .
Consequently, the mean time a PS resides in the system and in inventory is obtained by Little's law: T ( n ) = S ( n ) / eff 685
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Table 1. The expected profit for various values of n and . n\ 16 17 18 19 20 1 4.0455 4.0526 3.9930 3.8762 3.7099 2 4.4359 4.4397 4.3282 4.1208 3.8330 3 4.7169 4.7113 4.5564 4.2807 3.9071 4 4.9323 4.9172 4.7261 4.3955 3.9553 5 5.1057 5.0833 4.8618 4.4858 3.9913 6 5.2476 5.2203 4.9735 4.5589 4.0189 7 5.3639 5.3344 5.0667 4.6192 4.0403 8 5.4591 5.4299 5.1450 4.6692 4.0567 9 5.5363 5.5098 5.2110 4.7107 4.0689 10 5.5981 5.5764 5.2666 4.7451 4.0778 11 5.6465 5.6316 5.3134 4.7736 4.0838 12 5.6832 5.6767 5.3525 4.7970 4.0875 13 5.7097 5.7132 5.3851 4.8161 4.0890 14 5.7272 5.7419 5.4117 4.8313 4.0889 15 5.7367 5.7637 5.4332 4.8431 4.0872 16 5.7391 5.7794 5.4501 4.8521 4.0843 17 5.7352 5.7895 5.4629 4.8584 4.0802 18 5.7257 5.7947 5.4720 4.8624 4.0752 19 5.7111 5.7954 5.4777 4.8643 4.0694 20 5.6919 5.7919 5.4804 4.8644 4.0629
675
The following two directions for future research are proposed: (i) considering time deterioration of inventoried PSs; (ii) assuming customers arrival rates that depend on the inventory level of PSs. Acknowledgements This research was supported by the ISRAEL SCIENCE FOUNDATION (grant No. 1448/17). References Benjaafar, S., Cooper, W. L., Mardan, S. (2011). Production‐inventory systems with imperfect advance demand information and updating. Naval Research Logistics, 58(2), 88-106. Boxma, O. J., S. Schlegel, U. Yechiali. (2002). A note on the M/G/1 queue with a waiting server, timer and vacations. American Mathematical Society Translations, Series 2 207 25-35. Cox, D. R. (2017). The theory of stochastic processes. Routledge. Flapper, S. D. P., Gayon, J. P., Vercraene, S. (2012). Control of a production–inventory system with returns under imperfect advance return information. European Journal of Operational Research, 218(2), 392-400. Guha, D., V. Goswami, A. D. Banik. (2016). Algorithmic computation of steady-state probabilities in an almost observable GI/M/c queue with or without vacations under state dependent balking and reneging. Applied Mathematical Modelling 40(5) 4199-4219. Guo, P., & Hassin, R. (2011). Strategic behavior and social optimization in Markovian vacation queues. Operations research 59(4) 986-997. Guo, P., Z.G. Zhang. (2013). Strategic queueing behavior and its impact on system performance in service systems with the congestion-based staffing policy. Manufacturing & Service Operations Management 15(1) 118-131. Gupta, S. K. (1967). Queues with hyper-Poisson input and exponential service time distribution with state dependent arrival and service rates. Operations Research 15(5) 847-856. Hanukov, G., Avinadav, T., Chernonog, T., Spiegel, U., Yechiali, U. (2017). A queueing system with decomposed service and inventoried preliminary services. Applied Mathematical Modelling, 47, 276-293. Hanukov, G., Avinadav, T., Chernonog, T., Spiegel, U., Yechiali, U. (2018a). Improving efficiency in service systems by performing and storing “preliminary services”. International Journal of Production Economics, 197, 174-185. Hanukov, G., Avinadav, T., Chernonog, T., Yechiali, U. (2018b). Performance Improvement of a Service System via Stocking Perishable Preliminary Services. European Journal of Operational Research. Hwang, J., L. Gao, W. Jang. (2010). Joint demand and capacity management in a restaurant system. European Journal of Operational Research 207(1) 465-472. Iravani, S. M., Kolfal, B., Van Oyen, M. P. (2011). Capability flexibility: a decision support methodology for parallel service and manufacturing systems with flexible servers. IIE Transactions, 43(5), 363-382. Koshy, T., (2008). Catalan Numbers with Applications (Oxford University Press, Oxford). Latouche, G., Ramaswami, V., (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. Latouche, G., V. Ramaswami. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia, PA . Levy, Y., U. Yechiali. (1975). Utilization of idle time in an M/G/1 queueing system. Management Science 22 202-211. Levy, Y., U. Yechiali. (1976). An M/M/s queue with servers’ vacations. INFOR 14 153-163. Harchol-Balter, M. (2013). Performance modeling and design of computer systems: queueing theory in action. Cambridge University Press. Mytalas, G. C., M. A. Zazanis. (2015). An MX/G/1 queueing system with disasters and repairs under a multiple adapted vacation policy. Naval Research Logistics 62 171-189. Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Rosenberg, E., U. Yechiali. (1993). The MX/G/1 queue with single and multiple vacations under the LIFO service regime. Operations Research Letters 14(3) 171-179.
Table 1 shows that the optimal PS capacity and enhanced customers' arrival rate are n = 19 and = 17 , and that this point is the unique optimal solution of the expected profit maximization problem over the domain n, 20 . 6. CONCLUSIONS This paper analyzes an M/M/2-type system, which utilizes the server’s idle time to produce preliminary services for incoming customers. In addition, the server can publicize the number of available PSs (i.e. make the stock observable) which stimulates demand due to shorter sojourn time. In order to investigate such a system, a three-dimensional statespace is constructed, where the variables are (i) number of customers, (ii) inventory level of PSs and (iii) number of CSs. For a Markovian process and by applying matrix geometric analysis, closed-form expressions for the steady-state probabilities of the system states are derived, leading to the calculation of various performance measures. Due to the structure of the process and regardless of the size of the ratematrix R, it is possible in this model to derive closed-form expressions of all the entries of R. The relation of these expressions to Catalan numbers is shown. This result is notable in light of the fact that, in typical applications of the matrix geometric analysis, explicit calculation of the entries of R is rarely possible. Furthermore, it is proved analytically that the stability condition of our model is identical to that of a standard M/M/2 queue. In order to provide economic analysis, an optimization problem is formulated, where the objective is to maximize the server's expected profit by controlling the capacity of preliminary services and the investment in increasing customers arrival rate when preliminary services are available. Numerical examples reveal that the investigated profit function is quasi-concave.
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0 0 0 0 0 0 0 0 0 0 0 0 B2,3 = 0 0 0 0 0 0 0 0 0 0 0 0 ( 2 n +1)2 n
Urban, T. L. (2005). Inventory models with inventory-level-dependent demand: A comprehensive review and unifying theory. European Journal of Operational Research, 162(3) 792-804. Winston, W. (1978). Optimality of monotonic policies for multiple-server exponential queuing systems with state-dependent arrival rates. Operations Research 26(6) 1089-1094. Yang, D. Y., C. H. Wu. (2015). Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns. Computers & Industrial Engineering 82 151-158. Yechiali, U. (2004). On the MX/G/1 queue with a waiting server and vacations. Sankhya 66 159-174. Zhao, N., Lian, Z. (2011). A queueing-inventory system with two classes of customers. International Journal of Production Economics, 129(1), 225231.
Let I be the unit matrix. The matrix
Bi,2 = − ( a1 , a2 ,..., a2n ) I 2n2n , where,
APPENDIX − ( + 2 ) 2 0 0 2 2 − + ( ) B1,1 = − ( + 2 ) 0 0 0 0 0
B1, 2
B2,1
0 0 = 0 0 0 0 0 = 0 0 0 0
+ 2 + + ak = + 2 + + + 2
0 0 2 − ( n +1)( n +1)
k =1 k = 2,3,5,7,...,2i − 5 . k = 4,6,8,...,2i − 4,2i − 2 k = 2i − 3,2i − 1,2i + 1,...,2n − 1 k = 2i,2i + 2,...,2n
Similarly, 0 0 0 0 0 0
0 0
0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0
0 0
0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 ( n +1)( 2 n +1)
Bi,3 = ( b1 , b2 ,..., b2n ) I 2n2n , where
bk =
− ( + + ) 0 − ( 0 0 0 = 0 0 0 0
0 0 0 0 0 0 0 0 ( 2 n +1)( n +1)
0 0 + +) 0 − ( + + ) 0 0 − ( 0 0 0 0 0 0 0 0 0 0
k = 1, 2,3..., 2i − 5, 2i − 4, 2i − 2 . k = 2i − 3, 2i − 1, 2i, 2i + 1,..., 2n
A0 , A1 and A2 are given by A0 = I 2n2n ,
A1 = − ( a1 , a2 ,..., a2n ) I 2n2n , where
+ 2 ak = + + + 2
k =1 k = 2,3,5, 7,..., 2n − 1 , k = 4, 6,8,..., 2n
and A2 = [bi, j ]2n2n , where
bi , j
B2, 2
Bi , 2 is given by
0
i = j =1 2 i = j = 2 and i = j − 1 = 3,5, 7,..., 2n − 1 = i = j + 1 = 2,3 and i = j + 2 = 5, 7,9,..., 2n − 1 . 2 i = j + 2 = 4, 6,8,..., 2n Otherwise 0
0 0
0
0 0 0
0
0 0 0 0
0
+ +) 0 − ( 0 0 0 0
+ +) 0 − ( 0 0 0
+ +) 0 − ( 0 0
687
0
+ +) 0 0
, 0 − ( + ) − ( + ) ( 2 n+1)( 2 n+1) 0 0 0 0 0 0 0
0 0 0 0 0 0
ScienceDirect
IFAC PapersOnLine 52-13 (2019) 671–676
A multi-server queueing-inventory system with stock-dependent demand A multi-server queueing-inventory system with stock-dependent demand A multi-server queueing-inventory system with stock-dependent demand A multi-server queueing-inventory system with stock-dependent demand 1 2 system with stock-dependent 3 4 A multi-server queueing-inventory demand
Gabi Gabi Hanukov Hanukov11,, Tal Tal Avinadav Avinadav22,, Tatyana Tatyana Chernonog Chernonog33,, Uri Uri Yechiali Yechiali44 Gabi Hanukov11, Tal Avinadav22, Tatyana Chernonog33, Uri Yechiali44 Avinadav2, Tatyana Chernonog3, Uri Yechiali4 Gabi Hanukov 1 1, TalUniversity, Bar-Ilan Gan,Chernonog 5290002, Israel, [email protected] 11Department of Management, Gabi Hanukov , TalUniversity, Avinadav Ramat , Tatyana , Uri Yechiali of Management, Bar-Ilan Ramat Gan, 5290002, Israel, [email protected] 1Department 2 of of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] 22Department 1Department Department of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] Department of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] 2 3 1Department ofManagement, Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] Department of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] of Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] 33Department 2 Department of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] Department of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] 2 3 4 Department ofand Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] of Research, School of Sciences, Tel Aviv 4 3Department 4Department of Statistics Statistics and Operations Operations Research, SchoolRamat of Mathematical Mathematical Sciences, [email protected] Aviv University, University, Tel Tel Aviv, Aviv, 6997801, 6997801, of Management, Bar-Ilan University, Gan, 5290002, Israel, 3Department 4Department Department of Management, Bar-Ilan University, Ramat Gan, 5290002, Israel, [email protected] Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel, [email protected] 4 Israel, [email protected] School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 6997801, 4Department of Statistics and Operations Research, Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel, [email protected] Israel, [email protected] Israel, [email protected] Abstract: Abstract: We We consider consider aa two-server two-server service service system system in in which which idle idle servers servers produce produce and and store store preliminary preliminary Abstract: We consider a sojourn two-server service system customers in which idle servers produce and store preliminary services for reducing the time of incoming and increasing customers arrival rate. services forWe reducing thea sojourn timeservice of incoming and servers increasing customers arrival rate. We We Abstract: consider two-server system customers in which idle produce and store preliminary Abstract: We consider a sojourn two-server service system inawhich idle servers produce and store preliminary services forsystem reducing the time of incoming customers andbased increasing customers arrival rate. We model the as a Markovian process and provide method, on matrix geometric (MG) analymodel theforsystem as athe Markovian process and provide a method, on matrix geometric (MG) analyservices reducing sojourn time of incoming customers andbased increasing customers arrival rate. We services forsystem reducing sojourn time of incoming customers andbased increasing customers arrival rate. We model the as athe Markovian process and provide aprobabilities method, onrelevant matrix geometric (MG) analysis, to obtain closed-form solutions to the steady state and performance measures. sis, to the obtain closed-form solutions to the and steady state aprobabilities andonrelevant measures. model system as a Markovian process provide method, based matrix performance geometric (MG) analymodel the system as aofMarkovian process and provide aof method, based onrelevant matrix geometric (MG) analysis, to obtain closed-form solutions to the steady state probabilities and performance measures. We show the relation the elements in the rate matrix the MG to Catalan numbers, and prove that We to show the relation of thesolutions elementstointhe thesteady rate matrix the MG to and Catalan numbers, and prove that the the sis, obtain closed-form state of probabilities relevant performance measures. sis, obtain closed-form toin the state of probabilities relevant performance measures. We to show the relation of thesolutions elements thesteady rate matrix the MG to and Catalan numbers, and prove that the stability condition remains as is in the traditional M/M/2 queue, although the expected sojourn time of stability remains is in the in traditional M/M/2of queue, although the expected timethat of cuscusWe showcondition the relation of theaselements the rate matrix the MG to Catalan numbers,sojourn and prove the We show the relation of the elements in the rate matrix of the MG to Catalan numbers, and prove that the stability condition remains as is in the traditional M/M/2 queue, although the expected sojourn time of customers been provide an analysis for aa system in the PS tomers has has been reduced. reduced. We provide an economic economic analysis for although system the in which which the sojourn PS capacity capacity and the stability condition remainsWe as is in the traditional M/M/2 queue, expected time and of cusstabilityhas condition remains as is inarrival the traditional M/M/2 queue, the expected time and of customers been reduced. We provide an rate economic analysis for although a system in which the sojourn PS capacity the investment to increase customers' are decision variables. Copyright © 2019 IFAC investment to increase customers' arrival are decision variables. Copyright © 2019 IFAC tomers has been reduced. We provide an rate economic analysis for a system in which the PS capacity and the tomers has been reduced. We provide an rate economic analysis for a system in which the PS capacity and the investment to increase customers' arrival are decision variables. Copyright © 2019 IFAC investment to increase customers' arrival are system, decision variables. ©Ltd. 2019 © 2019, IFAC (International Federation of rate Automatic Control) Hosting Copyright by Elsevier AllIFAC rights reserved. Keywords: Markov process, queueing-inventory stock-dependent demand, matrix geometric investment increase customers' arrival rate are system, decision variables. Copyright © 2019 IFAC Keywords: to Markov process, queueing-inventory stock-dependent demand, matrix geometric analanalKeywords: Markov process, queueing-inventory system, stock-dependent demand, matrix geometric analysis, economic analysis ysis, economic analysis Markov Keywords: process, queueing-inventory system, stock-dependent demand, matrix geometric analKeywords: Markov process, queueing-inventory system, stock-dependent demand, matrix geometric analysis, economic analysis ysis, economic analysis Urban ysis, economic analysis Urban 2005) 2005) that that customers customers buy buy more more units units when when the the level level of of Urban 2005) that customers buy more units when the level of 1. INTRODUCTION the presented inventory is higher due to marketing effects 1. INTRODUCTION the presented inventory is higher dueunits to marketing effects buy more when the level of Urban 2005) that customers Urban 2005) that customers buy more when the level of 1. INTRODUCTION the presented inventory is higher dueunits to marketing effects (e.g., a wider selection). In our model, the stock of prelimi(e.g.,presented a wider selection). In higher our model, stock of prelimithe inventory is due the to marketing effects 1. INTRODUCTION Reducing customers1.waiting times in service systems is a INTRODUCTION the presented inventory is higher due to marketing effects (e.g., a wider selection). In our model, the stock of preliminary services has an additional positive effect on customers Reducing customers waiting times in service systems is a (e.g., nary services an additional effect onofcustomers a wider has selection). In our positive model, the stock prelimiReducing objective, customers which waitingaims times in service systems is a (e.g., common to reduce costs associated a wider selection). our positive model, the stock ofcustomers preliminary services has an additional effect onthe demand since it implies aaInshorter sojourn time in system. common to in reduce costs associated Reducing objective, customers which waitingaims times service systems is a nary demand since it implies shorter sojourn time in the system. services has an additional positive effect on customers Reducing customers waiting times in service systems is a demand common objective, which aims tosystem reduce costs associated with customers sojourn time in the and increase their nary services has an additional positive effect on customers since it implies a shorter sojourn time in the system. To model this effect, we use state-dependent customers' arriwith customers sojourn timeaims in thetosystem increase their demand common objective, which reduceand costs associated To model this effect, we use state-dependent customers' arrisince it implies a shorter sojourn time in the system. common objective, which aims tosystem reduce costs associated with customers sojourn time in the and increase their satisfaction. Usually, it is achieved by using faster serves demand since it implies a shorter sojourn time in the system. To model this effect, we use state-dependent customers' arrival rate (Gupta 1967, Winston 1978, Guo and Hassin 2011, satisfaction. Usually, is achieved by using faster serves with customers sojournit time in the system and increase their To val rate (Gupta 1967, Winston 1978, Guo and Hassin 2011, model this effect, we use state-dependent customers' arriwith customers sojourn inand the Zhang system and increase their To satisfaction. Usually, it time isGuo achieved by using faster serves (e.g., Hwang et al. 2010, 2013) or by hiring model this effect, we use state-dependent customers' arrival rate (Gupta 1967, Winston 1978, Guo and Hassin 2011, Zhao and Lian 2011). (e.g., Hwang Usually, et al. 2010, and Zhang 2013)faster or by serves hiring val satisfaction. it isGuo achieved by using Zhaorate and(Gupta Lian 2011). 1967, Winston 1978, Guo and Hassin 2011, satisfaction. Usually, it isGuo achieved by using faster (e.g., Hwang et al. 2010, and Zhang 2013) orcostly. by serves hiring additional servers. However, these methods are Reval rate (Gupta 1967, Winston 1978, Guo and Hassin 2011, Zhao and Lian 2011). additional servers. methods are orcostly. Re- Zhao and Lian 2011). (e.g., Hwang et al. 2010, Guo these and Zhang 2013) by hiring However, (e.g., Hwang et al. 2010, Guo these and2018a,b) Zhang 2013) orcostly. by hiring additional servers. However, methods aresuggested Recent (Hanukov et have to Our is and Lian 2011). motivated cent works works servers. (Hanukov et al. al. 2017, 2017, 2018a,b) have to Zhao Our model model is mostly mostly motivated by by the the fast fast food food industry industry in in additional However, these methods aresuggested costly. Readditional servers. However, these methods are costly. Recent works (Hanukov et al. 2017, 2018a,b) have suggested to Our model issuch mostly motivated patties by the or fast foodpizza, industry in use a server's idle time to produce and stock preliminary serwhich food, as hamburger basic can use aworks server's which food,issuch as hamburger basic can be be cent (Hanukov ettoal.produce 2017, 2018a,b) have suggestedserto Our model mostly motivated patties by the or fast foodpizza, industry in idle time and stock preliminary cent (Hanukov et 2017, 2018a,b) suggested to Our model issuch mostly motivated by the fast foodthe industry in as hamburger patties or basic pizza, can be use aworks server's idle time toal.produce and stockhave preliminary serwhich food, vices, which will be used to serve faster future customers, prepared before demand occurs, and only upon arrival of vices, which will be used to serveand faster customers, prepared before occurs, patties and only upon the arrival of use a server's idle time to produce stockfuture preliminary ser- which food, suchdemand as hamburger or basic pizza, can be use athus server's idle the time to produce and stock preliminary ser- prepared which food, suchdemand as heated hamburger patties oraccording basic pizza, can be before occurs, and only upon the arrival of vices, which will be used to serve faster future customers, and increase productivity of the service system witha customer they are up, adjusted to specific and thus increase of the service with- aprepared customer they demand are heated up, adjusted specific vices, which will the be productivity used to serve faster futuresystem customers, before occurs, and onlyaccording upon thetoarrival of customer they demand are heated up, adjusted according specific vices, which will be productivity used to serve faster futuresystem customers, before occurs, andtoonly upon thetoarrival of and thus increase the of the service with- aprepared out in or resources. innovaand served customer. out investing investing in additional additional or better better resources. This innovarequirement, andup, served to the the customer. Another and thus increase the productivity of the service This system with- customers acustomers customerrequirement, they are heated adjusted according to Another specific and investing thus increase the productivity of the service This system with- customers a customer they are heated adjusted according toretailing; specific requirement, andup, served to the customer. Another out in additional or better resources. innovative approach is modelled using a queueing-inventory system domain in which our model is applicable is bicycle tive investing approach in is modelled a queueing-inventory system customers domain in requirement, which our model applicable bicycle retailing; out additionalusing or better resources. This innovaand is served to the iscustomer. Another out investing additional or better resources. This(see innovacustomers and is served to the is Another domain in requirement, which our model applicable bicycle retailing; tive approachbyin is modelled using a queueing-inventory system represented two-dimensional Markov e.g. can assemble parts aacustomer. before represented two-dimensional Markov process process (see e.g. where where mechanics mechanics canmodel assemble parts of of is before aa tive approachbyisaamodelled using a queueing-inventory system domain in which our is applicable bicycle retailing; tive approach using a queueing-inventory system domainmechanics in which our model is applicable retailing; can assemble partsassemble of is a bicycle before a represented byisamodelled two-dimensional Markov process (see e.g. where Cox 2017), which is solved using a matrix geometric analycustomer's arrival, and subsequently the remaining Cox 2017), which is solved using aMarkov matrix process geometric customer's arrival, can and assemble subsequently the remaining represented by a two-dimensional (see e.g. where mechanics partsassemble of a bicycle before a analyrepresented byquestion a two-dimensional process e.g. customer's whereinmechanics assemble partsassemble of a bicycle before a Cox 2017), which is solved using aMarkov matrix geometric analyarrival, can and subsequently the remaining sis. The main in these models is how many (see prelimiwith the specific requirements parts in accordance accordance withsubsequently the customer’s customer’s specificthe requirements CoxThe 2017), which is solved using a matrix geometric analy- parts customer's arrival, and assemble remaining sis. main question in these models is how many prelimiparts in accordance with the customer’s specific requirements Cox 2017), which is solved using a matrix geometric analycustomer's arrival, and subsequently assemble the remaining sis. The main question in these models is how many preliminary services should be produced and stocked when the serv- and Handmade for are and preferences. preferences. Handmade nameplates for doors doorsrequirements are another another sis. The main should question these models is how when manythe prelimiin accordance with the nameplates customer’s specific nary services bein produced and stocked serv- parts sis.isThe main question these models iscontrast how when many prelimipartspreferences. in accordance with the customer’s specific requirements Handmade nameplates for doors arecan another nary services should beinproduced and stocked the serv- and er idle. Actually, this approach is in to extensive example in which service can be split up. The server proer is idle. Actually, this approach is in contrast to extensive example in which service can be split up. The server pronary services should be produced and stocked when the serv- and preferences. Handmade nameplates for doors arecan another nary shouldto bewhich produced and stocked when the servand preferences. Handmade doors arecan another example in which service cannameplates be splitceramic up.for The server proer is services idle.according Actually, this approach is in contrast to extensive literature servers' idle times may be used duce basic nameplates from wood, or glass (which literature according to which servers' idle times may be used duce basic nameplates from wood, ceramic or glass (which er is idle. Actually, this approach is in contrast to extensive example in which service can be split up. The server can proer is idle.according Actually, thiswhich approach is in contrast to extensive example in which service can wood, be splitceramic up. Theorserver probasic nameplates from glasscan (which literature to servers' idle times may be used duce for out (also as identical) before an made, the for carrying carrying out ancillary ancillary duties (also known as 'vacation 'vacation are not not identical) beforefrom an order order is ceramic made, and and complete the literature according to whichduties servers' idleknown times may be used are duce basic nameplates wood,is or complete glass (which literature according to which servers' idle timesRosenberg may be used ducenot basic nameplates wood, or complete glass (which are identical) beforefrom an order is ceramic made, and the for carrying out ancillary duties (also known as 'vacation models' in e.g. Levy and Yechiali 1975, 1976; and nameplate upon request (e.g., writing the name, add decoramodels' Rosenberg and are nameplate upon request writing the name, add decorafor carrying out ancillary duties 1975, (also 1976; known as 'vacation not identical) before (e.g., an order is made, and complete the in e.g. Levy and Yechiali for carrying ancillary (also known as Yang 'vacation nameplate are not identical)request before (e.g., an order is made, and complete the writing the name, add decoramodels' e.g.out Levy andetYechiali 1976; Rosenberg Yechiali Boxma al.duties 2002;1975, Yechiali 2004; Yechialiin 1993; Boxma Yang and tion, etc.). etc.). upon models' in1993; e.g. Levy andetYechiali 1976;2004; Rosenberg and tion, nameplate upon request (e.g., writing the name, add decoraal. 2002;1975, Yechiali etc.). upon request (e.g., writing the name, add decoramodels' in1993; e.g. Levy andet Yechiali 1975, 1976; Rosenberg and tion, nameplate Yechiali Boxma al. 2002; Yechiali 2004; Yang Wu 2015; Mytalas and Zazanis 2015; and Guha et al. 2016) Wu 2015;1993; Mytalas and et Zazanis 2015; and Guha al. 2016) Yechiali Boxma al. 2002; Yechiali 2004;et Yang and tion, etc.). 2. MODEL FORMULATION FORMULATION Yechiali 1993; Boxma al. 2002; Yechiali 2004;et Yang and tion, etc.). Wu 2015; Mytalas and et Zazanis 2015; andtimes. Guha al. 2016) which may increase waiting far, this 2. MODEL which mayMytalas increaseandcustomers customers waiting So far,2016) this Wu 2015; Zazanis 2015; andtimes. Guha So et al. 2. MODEL FORMULATION Assume a service system with two two servers, servers, in in which which fullfullWu 2015; Mytalas and Zazanis 2015; and Guha et al. 2016) which may increase customers waiting times. So far, this innovative approach has been applied (analytically) a service system FORMULATION with 2. MODEL innovative (analytically) only for which mayapproach increasehas customers waiting times. Soonly far, for thisaa Assume been applied 2. is MODEL FORMULATION Assume a service system with two servers, with in which fullwhich mayapproach increase customers waiting times.Poisson Soonly far,(custhisa service innovative has been applied (analytically) for service (FS) time exponentially-distributed mean 1/ single-server service systems and homogenous a service with two servers, with in which ,, (FS) time issystem exponentially-distributed mean full1/ single-server service has systems and homogenous Poisson innovative approach been applied (analytically) only (cusfor a Assume Assume a service witha Poisson two servers, in with which full(FS) time issystem exponentially-distributed with mean 1/ , innovative approach has been applied (analytically) only (cusfor a service single-server service systems and homogenous Poisson and customers arrival follows process rate tomer) arrival process. service (FS) time is exponentially-distributed withwith mean ,.. customers arrival follows a Poisson process rate1/ tomer) arrivalservice process.systems and homogenous Poisson (cus- and single-server service (FS) can time is split exponentially-distributed withwith mean ,. single-server and customers arrival follows a Poisson process rate1/ tomer) arrivalservice process.systems and homogenous Poisson (cus- The The service be into two phases: preliminary service service canarrival be split into two phases: process preliminary and customers follows a Poisson with service rate . tomer) arrival process. The current work applies the above innovative approach in and customers arrival follows a Poisson process with rate . tomer) arrival process. The service can be split into two phases: preliminary service (PS), which can be carried out without the presence of cusThe current work applies the above innovative approach in (PS), which can can be be split carried cusservice intoout twowithout phases:the preliminary service presence of The current work applies the above innovative approach in The the framework of a multi-server service system with stockThe service can be split into two phases: preliminary service (PS), which can be carried out without the presence of customers, and complementary service (CS), which can be given the framework The current work the above innovative in tomers, of aapplies multi-server service system approach with stockand complementary service (CS),the which can beofgiven which can be carried out without presence cusThe currentcustomers' work the above innovative in (PS), the framework of aapplies multi-server system approach with stockdependent arrival Multi-server are (PS), which be carried out without the ofgiven customers, and complementary service (CS), which can beis only when acan customer is present. present. When apresence server idle, dependent customers' arrival rate. rate.service Multi-server systems are only the framework of a multi-server service system systems with stockwhen a customer is When a server is idle, tomers, and complementary service (CS), which can be given the framework ofreal a multi-server system with stockdependent customers' arrival rate.service Multi-server systems are tomers, widely spread in life businesses, but in case of queueingand complementary service (CS), which can be given only when a customer is present. When a server is idle, he/she can produce PSs. In line with prior works in this dowidely spread in real life businesses, but in case of queueingdependent customers' arrival rate. Multi-server systems are he/she can produce PSs. is In present. line withWhen prior works in this dowhen a customer a server is idle, dependent customers' arrival rate. Multi-server systems are only widely spread in real life businesses, but in case of queueinginventory systems are considerably complicated to only when a customer is When a server is et idle, he/she can produce PSs. In present. line withet prior works in this domain (Benjaafar et al. 2011, Flapper al. 2012, Iravani al. inventory systems arelife considerably more complicated to anaana- main widely spread in real businesses,more but in case of queueinget al. 2011, Flapper al. 2012, al. he/she(Benjaafar can produce PSs. In line withetprior worksIravani in thisetdowidely spread in real businesses, butit in case of queueinginventory systems arelife considerably more complicated to analyze. As for stock-dependent demand, is known (see, e.g., he/she can produce PSs. In line with prior works in this domain (Benjaafar et 2011, Flapper et al. 2012, Iravani et al. 2012, Hanukov et al. 2017), the PS's preparation time is aslyze. As for stock-dependent demand, it is known (see, e.g., 2012, inventory systems are considerably more complicated to anaHanukov et al. 2017), the PS's preparation time is as(Benjaafar et 2011, Flapper et al. 2012, Iravani et al. inventory systems are considerably more complicated to analyze. As for stock-dependent demand, it is known (see, e.g., main main (Benjaafar et 2011, Flapper et al. 2012, Iravani et al. 2012, Hanukov et al. 2017), the PS's preparation time is aslyze. As for stock-dependent demand, it is known (see, e.g., 2012, Hanukov et al. 2017), the PS's preparation time is aslyze. As © for stock-dependent demand, it is known (see, Control) e.g., 682Hosting 2405-8963 IFAC (International Federation of Automatic by ElsevieretLtd. rightsthe reserved. 2012, Hanukov al. All 2017), PS's preparation time is asCopyright © 2019, 2019 IFAC Copyright 2019 responsibility IFAC 682Control. Peer review© of International Federation of Automatic Copyright ©under 2019 IFAC 682 10.1016/j.ifacol.2019.11.124 Copyright © 2019 IFAC 682 Copyright © 2019 IFAC 682
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Gabi Hanukov et al. / IFAC PapersOnLine 52-13 (2019) 671–676
sumed to be exponentially distributed with parameter α. The PSs are stored until the arrival of customers, and, as indicated, are used to reduce the sojourn time of future customers.
steady-state probabilities (i.e., pi, j ,k when the system is stable). Then, various performance measures, such as mean queue size, mean sojourn time, mean inventory level of PSs, mean time a PS spends in inventory, etc. are calculated. To do so, matrix geometric analysis (Neuts 1981) is applied, which entails calculation of the so-called rate matrix R (see bellow). The system’s states are arranged in the following lexicographic order (i = 2,3,…):
The total number of PSs in stock is limited to n (capacity), and when the inventoried PSs reach this capacity limit, the servers stop producing PSs and return to idle position. When a customer reaches the head of the line and a PS is available, his/her CS phase starts immediately. The CS time is exponentially distributed with mean 1/, where >. That is, the CS duration is stochastically shorter than an FS duration (first order stochastic dominance). When PSs are available, customers’ arrival rate increases to ( ) until the number of customers is equal to the number of PSs (which are both observable by the customers).
{( 0, 0, 0 ) , ( 0,1, 0 ) ,..., ( 0, n, 0 ) ; (1, 0, 0 ) , (1,1, 0 ) , (1,1,1) ,..., (1, n, 0 ) , (1, n,1) ;...;
( i, 0, 0 ) , ( i,1,1) , ( i, 2,1) , ( i, 2, 2 ) ,..., ( i, n,1) , ( i, n, 2 ) ;...}, and construct its infinitesimal generator matrix, denoted by Q: B1,1 B1,2 B2,1 B2,2 B3,1 0 0 0 Q= 0 0 0 0 0 0
The process can be formulated as a three-dimensional continuous-time Markov chain with a state space Lt , St ,Ut , where Lt 0,1, 2,..., denotes the number of customers in the system at time t, St 0,1, 2,..n denotes the total number of PSs at time t, and Ut 0,1, 2 denotes the number of customers at their CS phase at time t. Let L limt → Lt , and Let S limt → St U limt → Ut . pi, j ,k = Pr( L = i, S = j,U = k ) denote the joint probability
0
0
0
0
0
0
0
B2,3 B3,2
0
0
0
0
0
0
B3,3 B4,2
0
0
0
0
0
A2
B4,3
0
0
0
0
0
0
Bn +1,3 A1 A2
0
0 0
Bn +1,2 A2 0
0
0 0
A2 0 0
A0 A1
0 A0
, (1)
where matrices Bi, j and Ak ( k = 0,1, 2 ) are given in the Appendix.
distribution of the latter three-dimensional process. pi, j ,k
3. STEADY STATE ANALYSIS We now claim that the condition for stability of the system is identical to that of a regular M/M/2 queue without PSs (i.e., a Markovian service system with two parallel identical servers; see, e.g., Harchol-Balter 2013 p. 258): Theorem 1. The stability condition of the queuing-inventory system is 2 . Proof. The condition for stability of our system, according to Neuts (1981, p. 83) is (2) A0e A2e ,
represents the long run fraction of time that the system stays in state ( L = i, S = j,U = k ) . As an illustration, Figure 1 bellow depicts the system's states and its corresponding transition rate diagram for the case n = 3 . Each circle in the diagram depicts a state ( L = i, S = j,U = k ) and the arrows indicate the transition directions and rates.
where A0 and A2 (as well as A1 ) are given in the Appendix.
e = (1,1,...,1)T is a vector of ones, and = (1 , 2 ,..., 2n ) is the unique solution of the linear system A=0 (3) e = 1, where A A0 + A1 + A2 . In our case, = (1,0,...,0) . Thus, using A0 and A2 , the stability condition in equation (2) translates into 2 . Theorem 1 implies that although preparing PSs improves the system’s performance, it does not stringent its stability condition. This result can be further explained as follows: due to the stochastic nature of the system, at some point of time both servers will be fully occupied, and will not be able to produce any PS, so the system will become a regular M/M/2 queue. This explanation leads us to state the following conjecture:
Figure 1. System states and transition-rate diagram for n = 3.
Conjecture 1. The condition for stability of the queueinginventory system as described above having s identical servers is s .
Our goal is to obtain the system's stability condition (i.e., the condition on the parameters insuring that the queue size will not grow beyond any bound) and to derive the system’s 683
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0 C0.5( i − j + 3) 0.5( i − j +1) 0.5( i − j + 5) i − j +3 ( + 2 ) ( + + ) k +1 0.5(i − j −1) r i , j + 2 + 2 k C k ( 2 ) + 2k +2 ri , j = ( + 2 ) k =0 0.5( i − j −1) 0.5 ( i − j +1) − k r j + 1+ 2 k , j + 2 C 0.5 ( i − j −1) − k ( ) +2 i ( + 2 )( + + ) − j − 2 k k =0 2 0.5( i − j ) C0.5(i − j ) 0.5(i − j ) 0.5(i − j + 2) ( + 2 ) i − j +1
The matrix geometric analysis is based on derivation of a socalled rate-matrix R [ri, j ]2n2n , which is obtained by solving the quadratic matrix equation (see Neuts 1981) A0 + RA1 + R2 A2 = 02n2n .
(4)
Define the (n + 1) -dimensional vector of probabilities
(
p0 p0,0,0
p0,1,0
p0,2,0
)
p0,3,0
p0, n,0 ,
the (2n + 1) -dimensional vector of probabilities
(
p1 p1,0,0 p1,1,0 p1,1,1 p1,2,0 p1,2,1 p1,3,0 p1,3,1
p1, n,0
p1,n,1
)
and the (2n) -dimensional vectors of probabilities
(
pi pi ,0,0
pi ,1,1
pi ,2,1
pi ,2,2
pi,3,1
pi,3,2
pi, n,1
pi, n,2
i = n + 1, n + 2, n + 3,..., .
(5)
numerical calculations (see Chapter 8 in Latouche and Ramaswami 1999), mostly using successive substitutions (Neuts 1981, p. 37). However, for the current model, it is possible to derive closed-form expressions for all ri , j , as given in Theorem 2, due to the structure of the stochastic process given in Figure 1. ( 2m ) ! , Let Cm be the m-th Catalan number (i.e., Cm ( m + 1)!m! see Koshy 2008), where m is a non-negative integer. Then, Theorem 2. The entries of matrix R are: /(2 )
i =1 r22, 2 i=2 2 (1 − r2, 2 ) 0.5 ( i −1) ( + + ) ri , 2 + 2 k (r2 + 2 k , 2 + 2r2 + 2 k ,1 ) k =0 0.5(i −3) 0.5 ( i −1) − k r C ( ) 3 + 2 , 2 0 . 5 ( − 3 ) − k i k + − 3 − 2 i k k =0 i = 3,5,...,2n − 1 ( + + ) 0.5 ( i −1) − k 0.5 ( i − 5 ) r3+ 2 k ,1C0.5(i −3) − k ( ) ri ,1 = + 2 −1 ( + + ) i − 3 − 2 k k =0 2 ( + ) ( + 2 )r (2r + r ) 2 ,1 2, 2 i,2 0.5 ( i − 2 ) − k 0.5 ( i − 4 ) r C ( 2 ) + 0.5 4 + 2 k , 2 0.5( i − 4) − k i − 4 − 2 k ( + 2 ) k =0 0.5 ( i − 2 ) − k 0.5 ( i − 6 ) r4 + 2 k ,1C0.5(i − 4 ) − k (2 ) −1 i −4−2k ( 2 ) + k =0 i = 4,6,...,2n 4
ri , j
i = j − 1, j + 1,...,2n − 1 i = j , j + 2,...,2n
0 + + − ( + + ) 2 − 4 2 0.5 ( i −1) 0.5 ( i +1) 0.5 ( i − 5 ) r C ( ) 0.5(i −1) − k C0.5(i −1) + −1 3+ 2 k , 2 0.5(i −3) − k i −3− 2 k ( + + ) i − 2 + + ( ) k =0 0.5 ( i − 3) r3+ 2 k , 4 C0.5( i −3) − k ( ) 0.5(i −1) − k + 2 ( + + ) i − 3 − 2 k k =0 ri , 2 = k +1 0.5 ( i − 3) ri , 4 + 2 k ( + + )(C k (2 ) + r4 + 2 k , 2 ( + 2 ) 2 k +1 ) + ( + 2 ) 2 k +1 k =0 ( + + )(( + + ) − r2, 2 ) − 2 0.5( i − 2 ) C0.5(i − 2 ) 0.5( i − 2 ) 0.5i ( + 2 ) i − 3 0.5 ( i − 2 ) − k 0. 5 ( i − 6 ) r C ( 2 ) + k i − − k 4 2 , 2 0 . 5 ( 4 ) − 1 + (2 ) ( + 2 ) i − 4 − 2 k k =0 ( + 2 )(( + + ) − r2, 2 ) −
In most cases, the entries ri , j of the matrix R are found by
0 = 0.5( i − j ) 0.5( i − j + 2 ) C0.5( i − j ) i − j +1 ( ) + +
i = 1,2,..., j − 2
for j = 4,6,..., 2n .
)
i = 2,3, 4,... . Following Neuts (1981), the vectors pi satisfy i − n +1 pi = pn +1 R ( ) ,
673
i =1 i=2
i = 3,5,...,2n − 1
i = 4,6,...,2n
Proof. The solution has been obtained using induction, and can be verified by substitution in (5). As for calculating pi , i = 0,1,..., n + 1 , we use a subset of
equations from pQ = 0 (where p ( p0 , p1 , p2 ,...) ) combined
with
the
normalization
equa-
tion p0 e + p1 e +
p e = 1 . i
i =2
Consequently, we solve the following linear set: p0 B1,1 + p1 B2,1 = 0
p0 B1,2 + p1 B2,2 + p2 B3,1 = 0 pi −1 Bi ,3 + pi Bi +1,2 + pi +1 A2 = 0
i = 2,3,..., n
pn Bn +1,3 + pn +1 ( A1 + RA2 ) = 0 n
p0 e + p1 e +
p e + p i
n +1
I − R −1 e = 1.
i =2
i = 1, 2,..., j − 1 and i = j + 1, j + 3,..., 2n
4. PERFORMANCE MEASURES
i = j , j + 2,..., 2n − 1
Based on the steady state probabilities, we show how to calculate relevant systems performance measures. For a given capacity of n PSs, let L(n) and Lq (n) be the mean number
for j = 3,5,..., 2n −1 .
of customers in the system and in queue, respectively. Simi684
2019 IFAC MIM 674 Berlin, Germany, August 28-30, 2019
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and Tq ( n ) = Sq ( n ) / eff , respectively.
larly, let W (n) and Wq (n) be the mean sojourn time of a customer in the system and in queue, respectively; S (n) and Sq (n) be the mean number of PSs in the system and in in-
5. ECONOMIC ANALYSIS In this section, we provide an economic analysis of our queueing-inventory model. Let p be the revenue from supplying a service to a customer, so that the expected system's revenue rate is equal to peff . Let c be the sojourn cost per unit
ventory, respectively; and T (n) and Tq (n) be the mean time a PS resides in the system and in inventory, respectively.
Using
(4)
and
the
R
equations
i
−1
= I − R
and
−2
) e . (6)
of time per customer in the system, h—the holding cost per unit of time per inventoried PS (assuming no holding cost for a PS that moves on to the CS phase). Assume that the server can publicize the number of available PSs (i.e. make the stock observable) which stimulates demand due to shorter sojourn time. The publication may be generated in different levels. For example, the server can rent screen times for promotion in different locations near its store, where each screen has a potential to increase the number of potential customers. Another example is transmitting promotion messages to customers' smartphones in a larger distance (radius) to reach more customers. Thus, we denote by A( ) the cost rate of making the number of PSs observable (hereafter transparency cost) as a function of the desired arrival rate . We assume that the transparency cost is a convex increasing function of the gap in the arrival rate. Specifically, we use A( ) = a( − ) p , where a 0 and 1 . The objective is to maximize the total expected profit per time unit by controlling the PS capacity n and . Thus, the profit function is defined as follows
i =0
(i + 1) R
i
−2
= I − R , we obtain:
i =0
L ( n ) = p1e +
( n I − R
n
ip e + p
−1
n +1
i
i =2
+ I − R
Lq ( n ) = L ( n ) − 2 (1 − p0e ) + p1e .
(7)
In order to calculate the mean sojourn time of customer in system and in queue, we first have to calculate the effective customer's arrival rate eff . Let p be the proportion of time when customers arrive with rate , and p = 1 − p the proportion of time when customers arrive with rate . Then, the effective customers' arrival rate is obtained as follows (8) eff = p + p = ( − ) p + , where 1
p =
n
n
pi , j ,0 +
i = 0 j =1
n
p1, j ,1 +
j =2
n −1
n
pi , j ,1 +
i = 2 j =i
n
p
i , j ,2 .
(9)
i = 2 j =i +1
max
Then, W ( n ) = L ( n ) / eff and Wq ( n ) = Lq ( n ) / eff . In order to
obtain
S ( n) ,
we
vn +1 = ( 0 1 2 3
define
three
column
n0,1,2...
vectors
T
n n)
T
v2 n +1 = ( 0 1 1 2 2 3 3
(13)
To illustrate the behaviour of the expected profit as a function of PS capacity, n, and arrival rate when PSs are available, , we use the following parameter values: = 16 , = 10 , = 20 , = 18 , p = 0.5 , a = 0.2 , = 2 , c = 1 and h = 1 . The results are given in Table 1 and Figure 2.
n) ,
v2 n = ( 0 1 2 2 3 3
R(n, ) = peff − A( ) − cL(n) − hSq (n) .
and
n n ) . Thus, by using T
equation (4) and algebraic manipulations, we get
S ( n ) = p0 vn +1 + p1v2 n +1 +
n
p v
−1
i 2n
+ pn +1 I − R v2 n . (10)
i =2
Similarly, in order to get S q ( n ) , we define three column vectors
un +1 = ( 0 1 2 3
n) ,
n −1 n − 2)
and
u2 n = ( 0 0 1 0 2 1 3 2
T
u2 n +1 = ( 0 1 0 2 1 3 2
Sq ( n ) = p0un +1 + p1u2n +1 +
T
R(n,) 6
T
n
p u
i 2n
19
5
n n − 1) , so that
17
4
−1
+ pn +1 I − R u2 n . (11)
15
3
13 11
2
i =2
Given that inventory level is less than n, PSs are produced by the two servers when there are no customers in the system, and by one of the servers when there is only one customer in the system. Therefore, the effective production rate of PSs is eff = 2 ( p0 e − p0, n,0 ) + ( p1e − p1, n,0 − p1, n,1 ) . (12)
9
1
n
7 0
5
3
1
Figure 2. The expected profit as a function of n and .
Consequently, the mean time a PS resides in the system and in inventory is obtained by Little's law: T ( n ) = S ( n ) / eff 685
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Table 1. The expected profit for various values of n and . n\ 16 17 18 19 20 1 4.0455 4.0526 3.9930 3.8762 3.7099 2 4.4359 4.4397 4.3282 4.1208 3.8330 3 4.7169 4.7113 4.5564 4.2807 3.9071 4 4.9323 4.9172 4.7261 4.3955 3.9553 5 5.1057 5.0833 4.8618 4.4858 3.9913 6 5.2476 5.2203 4.9735 4.5589 4.0189 7 5.3639 5.3344 5.0667 4.6192 4.0403 8 5.4591 5.4299 5.1450 4.6692 4.0567 9 5.5363 5.5098 5.2110 4.7107 4.0689 10 5.5981 5.5764 5.2666 4.7451 4.0778 11 5.6465 5.6316 5.3134 4.7736 4.0838 12 5.6832 5.6767 5.3525 4.7970 4.0875 13 5.7097 5.7132 5.3851 4.8161 4.0890 14 5.7272 5.7419 5.4117 4.8313 4.0889 15 5.7367 5.7637 5.4332 4.8431 4.0872 16 5.7391 5.7794 5.4501 4.8521 4.0843 17 5.7352 5.7895 5.4629 4.8584 4.0802 18 5.7257 5.7947 5.4720 4.8624 4.0752 19 5.7111 5.7954 5.4777 4.8643 4.0694 20 5.6919 5.7919 5.4804 4.8644 4.0629
675
The following two directions for future research are proposed: (i) considering time deterioration of inventoried PSs; (ii) assuming customers arrival rates that depend on the inventory level of PSs. Acknowledgements This research was supported by the ISRAEL SCIENCE FOUNDATION (grant No. 1448/17). References Benjaafar, S., Cooper, W. L., Mardan, S. (2011). Production‐inventory systems with imperfect advance demand information and updating. Naval Research Logistics, 58(2), 88-106. Boxma, O. J., S. Schlegel, U. Yechiali. (2002). A note on the M/G/1 queue with a waiting server, timer and vacations. American Mathematical Society Translations, Series 2 207 25-35. Cox, D. R. (2017). The theory of stochastic processes. Routledge. Flapper, S. D. P., Gayon, J. P., Vercraene, S. (2012). Control of a production–inventory system with returns under imperfect advance return information. European Journal of Operational Research, 218(2), 392-400. Guha, D., V. Goswami, A. D. Banik. (2016). Algorithmic computation of steady-state probabilities in an almost observable GI/M/c queue with or without vacations under state dependent balking and reneging. Applied Mathematical Modelling 40(5) 4199-4219. Guo, P., & Hassin, R. (2011). Strategic behavior and social optimization in Markovian vacation queues. Operations research 59(4) 986-997. Guo, P., Z.G. Zhang. (2013). Strategic queueing behavior and its impact on system performance in service systems with the congestion-based staffing policy. Manufacturing & Service Operations Management 15(1) 118-131. Gupta, S. K. (1967). Queues with hyper-Poisson input and exponential service time distribution with state dependent arrival and service rates. Operations Research 15(5) 847-856. Hanukov, G., Avinadav, T., Chernonog, T., Spiegel, U., Yechiali, U. (2017). A queueing system with decomposed service and inventoried preliminary services. Applied Mathematical Modelling, 47, 276-293. Hanukov, G., Avinadav, T., Chernonog, T., Spiegel, U., Yechiali, U. (2018a). Improving efficiency in service systems by performing and storing “preliminary services”. International Journal of Production Economics, 197, 174-185. Hanukov, G., Avinadav, T., Chernonog, T., Yechiali, U. (2018b). Performance Improvement of a Service System via Stocking Perishable Preliminary Services. European Journal of Operational Research. Hwang, J., L. Gao, W. Jang. (2010). Joint demand and capacity management in a restaurant system. European Journal of Operational Research 207(1) 465-472. Iravani, S. M., Kolfal, B., Van Oyen, M. P. (2011). Capability flexibility: a decision support methodology for parallel service and manufacturing systems with flexible servers. IIE Transactions, 43(5), 363-382. Koshy, T., (2008). Catalan Numbers with Applications (Oxford University Press, Oxford). Latouche, G., Ramaswami, V., (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. Latouche, G., V. Ramaswami. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia, PA . Levy, Y., U. Yechiali. (1975). Utilization of idle time in an M/G/1 queueing system. Management Science 22 202-211. Levy, Y., U. Yechiali. (1976). An M/M/s queue with servers’ vacations. INFOR 14 153-163. Harchol-Balter, M. (2013). Performance modeling and design of computer systems: queueing theory in action. Cambridge University Press. Mytalas, G. C., M. A. Zazanis. (2015). An MX/G/1 queueing system with disasters and repairs under a multiple adapted vacation policy. Naval Research Logistics 62 171-189. Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Rosenberg, E., U. Yechiali. (1993). The MX/G/1 queue with single and multiple vacations under the LIFO service regime. Operations Research Letters 14(3) 171-179.
Table 1 shows that the optimal PS capacity and enhanced customers' arrival rate are n = 19 and = 17 , and that this point is the unique optimal solution of the expected profit maximization problem over the domain n, 20 . 6. CONCLUSIONS This paper analyzes an M/M/2-type system, which utilizes the server’s idle time to produce preliminary services for incoming customers. In addition, the server can publicize the number of available PSs (i.e. make the stock observable) which stimulates demand due to shorter sojourn time. In order to investigate such a system, a three-dimensional statespace is constructed, where the variables are (i) number of customers, (ii) inventory level of PSs and (iii) number of CSs. For a Markovian process and by applying matrix geometric analysis, closed-form expressions for the steady-state probabilities of the system states are derived, leading to the calculation of various performance measures. Due to the structure of the process and regardless of the size of the ratematrix R, it is possible in this model to derive closed-form expressions of all the entries of R. The relation of these expressions to Catalan numbers is shown. This result is notable in light of the fact that, in typical applications of the matrix geometric analysis, explicit calculation of the entries of R is rarely possible. Furthermore, it is proved analytically that the stability condition of our model is identical to that of a standard M/M/2 queue. In order to provide economic analysis, an optimization problem is formulated, where the objective is to maximize the server's expected profit by controlling the capacity of preliminary services and the investment in increasing customers arrival rate when preliminary services are available. Numerical examples reveal that the investigated profit function is quasi-concave.
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0 0 0 0 0 0 0 0 0 0 0 0 B2,3 = 0 0 0 0 0 0 0 0 0 0 0 0 ( 2 n +1)2 n
Urban, T. L. (2005). Inventory models with inventory-level-dependent demand: A comprehensive review and unifying theory. European Journal of Operational Research, 162(3) 792-804. Winston, W. (1978). Optimality of monotonic policies for multiple-server exponential queuing systems with state-dependent arrival rates. Operations Research 26(6) 1089-1094. Yang, D. Y., C. H. Wu. (2015). Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns. Computers & Industrial Engineering 82 151-158. Yechiali, U. (2004). On the MX/G/1 queue with a waiting server and vacations. Sankhya 66 159-174. Zhao, N., Lian, Z. (2011). A queueing-inventory system with two classes of customers. International Journal of Production Economics, 129(1), 225231.
Let I be the unit matrix. The matrix
Bi,2 = − ( a1 , a2 ,..., a2n ) I 2n2n , where,
APPENDIX − ( + 2 ) 2 0 0 2 2 − + ( ) B1,1 = − ( + 2 ) 0 0 0 0 0
B1, 2
B2,1
0 0 = 0 0 0 0 0 = 0 0 0 0
+ 2 + + ak = + 2 + + + 2
0 0 2 − ( n +1)( n +1)
k =1 k = 2,3,5,7,...,2i − 5 . k = 4,6,8,...,2i − 4,2i − 2 k = 2i − 3,2i − 1,2i + 1,...,2n − 1 k = 2i,2i + 2,...,2n
Similarly, 0 0 0 0 0 0
0 0
0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0
0 0
0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 ( n +1)( 2 n +1)
Bi,3 = ( b1 , b2 ,..., b2n ) I 2n2n , where
bk =
− ( + + ) 0 − ( 0 0 0 = 0 0 0 0
0 0 0 0 0 0 0 0 ( 2 n +1)( n +1)
0 0 + +) 0 − ( + + ) 0 0 − ( 0 0 0 0 0 0 0 0 0 0
k = 1, 2,3..., 2i − 5, 2i − 4, 2i − 2 . k = 2i − 3, 2i − 1, 2i, 2i + 1,..., 2n
A0 , A1 and A2 are given by A0 = I 2n2n ,
A1 = − ( a1 , a2 ,..., a2n ) I 2n2n , where
+ 2 ak = + + + 2
k =1 k = 2,3,5, 7,..., 2n − 1 , k = 4, 6,8,..., 2n
and A2 = [bi, j ]2n2n , where
bi , j
B2, 2
Bi , 2 is given by
0
i = j =1 2 i = j = 2 and i = j − 1 = 3,5, 7,..., 2n − 1 = i = j + 1 = 2,3 and i = j + 2 = 5, 7,9,..., 2n − 1 . 2 i = j + 2 = 4, 6,8,..., 2n Otherwise 0
0 0
0
0 0 0
0
0 0 0 0
0
+ +) 0 − ( 0 0 0 0
+ +) 0 − ( 0 0 0
+ +) 0 − ( 0 0
687
0
+ +) 0 0
, 0 − ( + ) − ( + ) ( 2 n+1)( 2 n+1) 0 0 0 0 0 0 0
0 0 0 0 0 0