Transient results for M/M/1/c queues via path counting. M. Hlynka*, L.M. Hurajt and M. Cylwa

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1 Int. J. Mathematcs n Operatonal Research, Vol. X, No. X, xxxx 1 Transent results for M/M/1/c queues va path countng M. Hlynka*, L.M. Hurajt and M. Cylwa Department of Mathematcs and Statstcs, Unversty of Wndsor, Wndsor, Ontaro N9B 3P4, Canada E-mal: hlynka@uwndsor.ca *Correspondng author Abstract: We fnd combnatorally the probablty of havng n customers n an M/M/1/c queueng system at an arbtrary tme t when the arrval rate λ and the servce rate μ are equal, ncludng the case c =. Our method uses pathcountng methods and fnds a bjecton between the paths of the type needed for the queueng model and paths of another type whch are easy to count. The bjecton nvolves some nterestng geometrc methods. Keywords: countng; M/M/1; M/M/1/c; paths; queueng; transent. Reference to ths paper should be made as follows: Hlynka, M., Hurajt, L.M. and Cylwa, M. (xxxx) Transent results for M/M/1/c queues va path countng, Int. J. Mathematcs n Operatonal Research, Vol. x, No. x, pp.xx xx. Bographcal notes: Myron Hlynka s a Professor n the Department of Mathematcs and Statstcs at the Unversty of Wndsor (Wndsor, Ontaro, Canada). Hs man research and publcatons are n the areas of queueng theory and appled probablty. He s the Webmaster of a popular Queueng Theory webste wth URL web.uwndsor.ca/math/hlynka/queue.html. Laura Hurajt s a graduate student n the Department of Mathematcs and Statstcs at the Unversty of Wndsor wth nterests n queueng theory and Le groups, and holds an Natural Scence and Engneerng Research Councl of Canada (NSERC) post graduate award. Mchelle Cylwa s an undergraduate student n the Department of Mathematcs and Statstcs at the Unversty of Wndsor wth nterests n queueng theory and stochastc processes, and held an NSERC Undergraduate Research award durng Summer, 008, when she dd research for ths artcle. 1 Introducton In queueng theory, the most commonly studed system s the M/M/1 system whch has exponentally dstrbuted nterarrval tmes, exponentally dstrbuted servce tmes and a sngle server. The lmtng probabltes, whch are easy to obtan, of fndng customers n the system (for = 01,,, ) form a geometrc dstrbuton and requre the arrval rate λ to be less than the servce rate μ. However n the transent case (fnte tme), the probabltes are most commonly derved usng dfferental equatons and are expressed Copyrght 00x Inderscence Enterprses Ltd.

2 M. Hlynka, L.M. Hurajt and M. Cylwa n terms of Bessel functons (Gross et al., 008). In the transent case, the restrcton that λ < μ s not necessary. Transent results n queueng are recevng consderable attenton n recent queueng lterature. In ths artcle, we consder a combnatoral method for fndng the transent probabltes at tme t when λ = μ. Successful combnatoral analyses for transent probabltes of an M / M /1 queue date back to Champernowne (1956) whle further achevements n combnatoral queueng are documented n Takacs (1967) and Jan, Mohanty and Bohm (007). The type of geometrc combnatoral argument presented here s new and qute appealng. It handles n a unfed way, M / M /1 and M / M/1/ c cases startng wth an ntal number of customers that can be zero or non-zero. The form of many of the results s also new. In Secton, we present a bref lterature revew. In Secton 3, we ndcate the relatonshp between transent queueng probabltes and path counts. In Secton 4, we fnd an nterestng geometrc transformaton whch ads n path countng for the M/M/1 case. In Secton 5, we fnd a geometrc transformaton whch ads n path countng n the M/M/1/c case. Some addtonal nterestng propertes arse n ths case. Conclusons and comments follow at the end. Lterature revew There s consderable current research on transent queueng models. In partcular, there s nterest n dfferent forms of solutons (to a varety of queueng problems) and dfferent ways of obtanng such solutons. Sharma (1997) obtans a consderable number of transent queueng results often usng Laplace transforms. Krnk and Mortensen (007) fnd transent solutons, va recurson, for brth death type models that allow catastrophes. Krshna Kumar and Pava Madheswar (005, 007) consder transent analyss of an M/M/1 queue wth catastrophes and server falures. Bohm, Krnk and Mohanty (1997) use combnatoral countng methods to fnd expressons for the transent probabltes for M / M /1 and M / M / c models. Grffths, Leonenko and Wllams (006) fnd expressons for transent probabltes of M / E k / 1 queueng models. Ther soluton uses dfferental equatons and generatng functons. Jan and Mete (005) remark on the varety of dfferent soluton forms that have been obtaned for transent probabltes n the M / M / 1 case. Taraba and El-Baz (006) obtan a relatvely smple expresson for the transent probabltes n the M / M / 1/ c case by assumng that the soluton has a partcular form. Combnatoral expressons show up n ther dervatons. Van Houdt and Blonda (005) consder transent queue length and watng tme dstrbutons for a dscrete tme queueng system. Margolus (007) consders transent solutons to quas-brth and death processes wth tme-varyng perodc rates. b Joy and Jones (005) use expressons for transent probabltes of M / M / 1 batch arrval queues n order to compute the probablty of servng at least k customers by tme t. Ths probablty s then proposed as one of several performance measures for hosptal servce.

3 Transent results for M/M/1/c queues va path countng 3 There are many other papers for transent probabltes n queues expressed n a varety of forms and derved n a varety of ways. 3 Queueng model In ths secton, we have gven an expresson for the number of customers n an M/M/1 queue at tme t n terms of certan path counts. Consder a sngle server M/M/1 queueng system wth arrvals at rate λ and servce at rate μ. Assume that at tme zero, there are zero customers n the system. We are nterested n the probablty of fndng customers n the system at tme t. If we want to smulate such a system after each arrval or servce completon, we could generate two exponental random values, one nterarrval tme and one completon tme, and choose the mnmum to determne what the next event (an arrval or a departure) would be. However, f there are zero customers n the system and the next event (mnmum) s a departure then we nterpret ths to mean that the system stays wth zero customers, but the clock moves forward by the mnmum of the two random values. The embedded Markov chan s that assocated wth the process wth respect to the rate λ + μ. Let p0, () t be the probablty that at tme t, there are customers n the system, gven that there were 0 customers at tme 0. Durng the tme length t, there wll be some number of events k (arrvals or departures wth the count havng a Posson dstrbuton). The k events wll result n customers present n the system at tme t f there s a path of k steps from (0, 0) to ( k, ) where the path has steps of type (*) below. (*) For m > 0, a step s a movement from an arbtrary feasble pont ( jm, ) to ( j+, 1 m+ 1) or from ( jm, ) to ( j+ 1, m 1). If m = 0, then a step s a movement from ( j, 0) to ( j +, 1 0) or from ( j, 0) to ( j + 11),. Theorem 1. For an M/M/1 queue, assume λ = μ. Then, p 0, k μt k μt ( μt) e 0, μ 0, k k! k! k= k= N ( k) ( t) e N ( k) () t = = (1) where N 0, (k) s the number of paths from (0, 0) to (k, ) n k steps. If λ μ, k k ( λ+ μ) t (( + ) t) e 0, 0,, f k k= f = 0 u d+ f λ μ λ μ p () t = N ( k) ()! λ+ μ λ+ μ where N0,, f( k) s the number of paths from (0, 0) to ( k, ) n k steps wth exactly f flat steps, where u, d can be solved n terms of k,, f from the restrctons u+ d + f = k, u d =. Proof. The Posson probablty of k events n tme t s k ( λ+ μ) t ((( λ+ μ) t) e ) / k! for k = 01,,. The probablty of movng up on any step s λ /( λ+ μ). If λ = μ then the probablty of movng up at each step s 1/. If the movement on a step s not up then t

4 4 M. Hlynka, L.M. Hurajt and M. Cylwa must be down (or flat f the start of a step s (, 0) ). To obtan Equaton (1), we condton on k and sum over all cases. If λ μ then the probabltes of two paths whch end at wll be equal f they have the same number of flat steps. So, we partton our summaton nto classes wth the same number of flat steps f. We condton on k and then on f and sum over all cases to get (). The man contrbuton of ths artcle s the study of N0, ( k ) (for both the M/M/1 and M/M/1/c cases), examnng the geometrc nterpretaton and fndng an expresson for t. It s nterestng to compare our result wth the standard expresson usng Bessel functons gven below (Gross et al., 008). 1 ( λ+ μ) t λ λ p0, () t = e I( at) + I+ 1( at) μ μ j λ λ λ + 1 I j ( at) μ μ μ j=+ 1/ where a = μλ ( / μ) and expresson reduces to I k ( y ) = (( y / ) ) / (( k + m )! m!). When λ = μ, ths k+ m m= 0 μt 0, μ + 1 p () t = e ( I ( t) + I ( μt)). 4 Path countng for M/M/1 queues In Secton 4.1, we fnd some results for path counts N ˆ 0, ( k ) whch are related to the path counts N0, ( k ) that we seek. In Secton 4., we gve a transformaton from one type of path to another. In Secton 4.3, we gve the nverse transformaton. 4.1 Prelmnares We begn wth standard notaton on paths and extend t to paths of the type we need. We also suggest the exstence of a bjecton that wll help to obtan our path counts. Defnton 1. An upward step U from ( ab, ) s a lne segment connectng the lattce ponts ( ab, ) and ( a+ 1, b+ 1). A downward step D s a lne segment connectng ( ab, ) to ( a+, 1b 1). A flat step F s a lne segment connectng ( a, 0) to ( a + 10),. In the followng defnton, a path allows negatve y coordnates. Defnton. An U-D path of length k s a polygonal path composed of a sequence of upward and downward steps only, begnnng at the orgn and termnatng at the lattce pont ( k, ) where k and are ntegers and k + 0 mod.

5 Transent results for M/M/1/c queues va path countng 5 Defnton 3. Let P be a polygonal path of length k composed of upward, downward and flat paths. If P s restrcted to the frst quadrant and flat paths only occur on the horzontal axs, then P s called a U-D-F path. Lemma 1. If k represents the length of a U-D path, wth 0 as the ntal heght, and as the termnal heght then the number of such paths, denoted N ˆ 0, ( k ), s gven by ˆ 0 ( ) k N, k = k +. (3) Proof. Let u and d represent the numbers of upward and downward steps, respectvely, contaned n an U-D path of length k termnatng at heght. Then, we have = u d and k = u+ d. Solvng for u gves u = ( k+ )/. Snce there are only two choces for each segment, upward or downward, the number of such paths s N k k ( k) = =. 0, k+ u Let S 1 be the set of all lattce ponts at whch a U-D path may termnate, that s, S1 = {( k, ) k > 0, k k, k+ 0mod}. If each such lattce pont s connected to ts neghbours only by an upward or a downward step and s assgned a label α where α s the number of U-D paths termnatng at that pont, then we have a representaton of Pascal s trangle (Fgure 1). Fgure 1 S 1 wth connectng steps

6 6 M. Hlynka, L.M. Hurajt and M. Cylwa Thus, for example, after step 6, we have N0, 6(6) N0, 4(6) N0, (6) N00, (6) N0, (6) N04, (6) N06, (6) Now suppose that S 1 s folded along the lne = 1/ (or Y = 1/ n the usual X-Y coordnate system), so that the each pont ( k, ) below the k -axs s mapped to the pont ( k, 1). The result s a set of ponts, S = {( k, ) k > 0, 0 k, k, Z } whch s the set of all lattce ponts at whch an U-D-F path may termnate. Fgure shows S connected usng the permssble steps. If each Pascal label α from S 1 s carred over to S, then α corresponds to the number of U-D-F paths termnatng at that pont. Thus for example, f N 0 (6), represents the number of U-D-F paths from 0 to after 6 steps, the counts can be shown to be N00, (6) N01, (6) N0, (6) N03, (6) N04, (6) N05, (6) N06, (6) Compare these values wth N 0 (6), together wth the foldng over = 1/. The two results are the same! Ths suggests that there may exst a bjectve functon that maps each U-D path to a unque U-D-F path n such a way that we can easly count the U-D-F paths. In fact, a U-D path can be transformed to the correspondng U-D-F path as shown below. Upward and downward steps are denoted as U and D. A sample path has been chosen to demonstrate the path transformaton algorthm. It s qute lkely that the bjecton between the two types of path can have other uses than the use gven here. Fgure S wth connectng steps

7 Transent results for M/M/1/c queues va path countng 7 Fgure 3 Convertng an U-D path nto U-D-F 4. U-D paths to U-D-F paths In ths secton, we ndcate the steps of one drecton of our bjectve map between two types of paths. 1 Separate the U-D path nto sectons such that the dvsons occur where the path touches or crosses the horzontal axs and dentfy the postve and negatve sectons (Fgure 3a). + U D D D U U + U U D D D U D D U D.

8 8 M. Hlynka, L.M. Hurajt and M. Cylwa Change each letter whch les wthn a negatve secton and adjacent to a dvson nto a flat path (Fgure 3b). + U D F D U F + U U D D F F F D U D. 3 Reverse the remanng letters n the negatve sectons (Fgure 3c). + U D F U D F + U U D D F F F U D U. 4.3 U-D-F paths to U-D paths In ths secton, we demonstrate the other drecton of the bjecton and ndcate the steps to move from U-D-F paths to U-D paths. 1 Identfy the flat paths and group them n pars from left to rght (Fgure 3c). U D [F U D F] U U D D [F F] [F U D U. Reverse the letters contaned between each par of flat paths (Fgure 3b). U D [F D U F] U U D D [F F] [F D U D. 3 Change the frst flat path of each par to a downward path and the second one to an upward path (Fgure 3a). U D [D D U U] U U D D [D U] [D D U D. Snce, we have demonstrated rules to convert paths back and forth, we have a bjecton between U-D-F paths and U-D paths. If the U-D path termnates at the pont ( k, ) where k 0 then the correspondng U-D-F path also termnates at ( k, ). However, an U-D path whch ends at the pont ( k, ) wth < 0 corresponds to a U-D-F path that ends at the pont ( k, 1). If nstead we begn wth an U-D-F path that ends at ( k, ) such that k + 0mod then the correspondng U-D path also termnates at the pont ( k, ). Smlarly, an U-D-F path whch termnates at ( k, ) such that k+ 1mod s transformed nto an U-D path that ends at the pont ( k, 1). 4.4 Countng U-D-F paths The bjecton that has been found n the prevous sectons allows us to count our U-D-F paths by transformng them to U-D paths whch are easy to count. Theorem. Let k represent the length of the U-D-F path and represent the heght at whch the path termnates. Then, the number of such U-D-F paths, denoted N0, ( k ), s gven by N k ( k) =, 0, k++ 1 where [ ] s the greatest nteger functon. (4)

9 Transent results for M/M/1/c queues va path countng 9 Proof. Snce, there exsts a bjecton between the U-D-F paths and the U-D paths, there are the same number of U-D-F paths as there are correspondng U-D paths. Usng Lemma 1, we have N 0, k k+ N0, ( k) f k + 0 mod ( k) = = N ( k) f k + 1 mod k 0, 1 k 1 f k + 0 mod f k + 1 mod. The property n n r n r For k + 1 mod, = s employed to smplfy the expresson. Snce N k k k ( k) = = =. 0, k 1 k 1 k++ 1 k k+ k f k+ 0 mod k++ 1 = f k + 1 mod, then n general, we have N k ( k) =. 0, k Fndng transent probabltes for M/M/1 queues Example 1. Suppose customers arrve at an M/M/1 queueng system at a rate of λ = 0 customers per hour. The servce tme s μ = 0 customers per hour. At tme 0, there are 0 customers. After 6 mn (0.1 hour), we the probablty of havng customers n the system for = 01,,,..., 5. Soluton. Snce, λ = μ we can use Equaton (1) from Theorem 1 together wth Equaton (4) of Theorem. Usng a computer algebra system and truncatng at k = 0, we fnd the probabltes (to fve decmal places) to be: p00, (0. 1) = p01, (0. 1) = p0, (0. 1) = p03, (0. 1) = p (0. 1) = p (0. 1) = , 05,

10 10 M. Hlynka, L.M. Hurajt and M. Cylwa 4.6 Recursve expresson for N0 ( k),, f We can also handle computatons for the transent queueng probabltes n the λ μ case usng Equaton (). Ths s done by buldng a recursve computatonal expresson for N0,, f( k) n Theorem 1 as follows. Theorem 3. Let N0,, f( k) be the number of paths from 0 to n k steps wth exactly f flat steps. Then, for 0, N ( k) = N ( k 1) + N ( k 1) 0,, f 0,+, 1 f 0,, 1 f and N ( k) = N ( k 1) + N ( k 1). 00,, f 01,, f 00,, f 1 However, ths method would be less convenent than usng the expressons presented n Sharma (1997), for example. 4.7 Countng U-D-F paths startng from (0, m) In an M/M/1 queueng system, we mght wsh to know the probablty of havng customers n the system f we start wth mm ( 0) customers at tme 0 where the arrval rate λ equals the servce rate μ. We wll see that our method treats the m = 0 and the m 0 case geometrcally n the same way. Let Nm, ( k) be the number of paths from (0, m) to ( k, ). The analogue to Theorem 1 s the followng. Theorem 4. Wth the condtons above, assume λ = μ. Let pm, () t be the probablty of havng customers n the system at tme t f there are m customers at tme 0. Then p m, k μt ( μt) e Nm, ( k) () t =. (5) k k! k = 0 The count of the number of paths Nm, ( k) uses the same foldng at = 1/ as was used prevously. For example, suppose we consder paths of length k = 6 startng from (0, m) for m =. The counts of the U-D paths are N, 4(6) N, (6) N0, (6) N, (6) N4, (6) N6, (6) N8, (6) Foldng over = 1/, the U-D-F counts become N00, (6) N01, (6) N0, (6) N03, (6) N04, (6) N06, (6) N08, (6) Thus, we get our bnomal counts, but n a strange order! The analogue of Theorem s the followng.

11 Transent results for M/M/1/c queues va path countng 11 Theorem 5. Consder all U-D-F paths from (0, m) to ( k, ). Then, the number of such U- D-F paths, denoted Nm, ( k), s gven by N m, k k+ m ( k) = k k++ m+1 f k + m 0 mod f k + m 1 mod. (6) 5 Path countng for M/M/1/c queues We next assume that we are dealng wth an M/M/1/c queue wth c <. We want to obtan an expresson for the probablty of havng customers n the system at fnte tme t. In ths case, there s a capacty of c customers n the system, ncludng the customer n servce. As n the M/M/1 case, we assume that each step conssts of an arrval or a departure. If there are 0 customers n the system, and the next event s a departure, then we assume that the clock moves forward, but the path s Flat. Smlarly, f there are c customers n the system and the next event s an arrval, then we assume that the clock moves forward but the path s Flat (at level c ). Therefore, we may have two dfferent types of flat path n an M / M / 1/ c system. Let p0, () t be the probablty that at tme t, there are customers n the system, gven that there were 0 customers at tme 0. Durng the tme length t, there wll be some number of events k (arrvals or departures wth the count havng a Posson dstrbuton). The k events wll result n customers present n the system at tme t f there s a path of k steps from (0, 0) to ( k, ) where the path has steps of type (*) below. (*) For an M / M / 1/ c queue wth 0 < m< c, a step s a movement from an arbtrary feasble pont ( jm, ) to ( j+, 1 m+ 1) or from ( jm, ) to ( j+ 1, m 1). If m = 0, then a step s a movement from ( j, 0) to ( j + 10), or from ( j, 0) to ( j + 1, 1). If m= c, then a step s a movement from ( jc, ) to ( j+, 1 c) or from ( jc, ) to ( j+ 1, c 1). Our analogue of Theorem 1 s Theorem 6. Consder an M / M / 1/ c queue, assume λ = μ. Then p 0, k μt k μt ( μt) e 0, μ 0, k k! k k! = k= M ( k) ( t) e M ( k) () t = = (7) where M0, ( k ) s the number of allowable paths from (0, 0) to ( k, ) n k steps. If λ μ, k k f k ( λ+ μ) t (( + ) t) e 0, 0,, f, g k k= f = 0 g= 0 u+ g d+ f λ μ λ μ p () t = M ( k)! λ+ μ λ+ μ (8)

12 1 M. Hlynka, L.M. Hurajt and M. Cylwa where M 0,, f, g( k) s the number of allowable paths from (0, 0) to ( k, ) n k steps wth exactly f flat steps at level 0, wth exactly g flat steps at level c, where u, d can be solved n terms of k,, f, g from the restrctons u+ d + f + g = k, u d =. k ( λ+ μ) t Proof. The Posson probablty of k events n tme t s ((( λ+ μ) t) e ) / k! for k = 01,,. The probablty of movng up (or flat from (, c) ) on any step s λ /( λ+ μ). If λ = μ, then the probablty of movng up at each step s 1/. The probablty of movng down (or flat from (, 0) ) on any step s μ /( λ+ μ). To obtan Equaton (7), we condton on k and sum over all cases. If λ μ, then the probabltes of two paths whch end at wll be equal f they have the same number of flat (level 0) steps and the same number of flat (level c ) steps. So we partton our summaton nto classes wth the same number of (0 level) flat steps f and the same number of ( c level) flat steps g. We condton on k, f and g and then sum over all cases to get Equaton (8). For an M / M / 1/ c queue, the possble paths for the number of customers n the system after k steps are referred to as U-D-F-F paths snce, there are two types of flat path possble. Let S 3 be the set of all lattce ponts at whch a U-D-F-F path may termnate, that s, S3 = {( k, ) 0< k, 0 mn{ k, c}}. We want to fnd a useful bjecton (one that helps us to count the paths) between U-D paths as descrbed n Secton 4 and U-D-F-F paths. In Fgure 4, we llustrate the possble U-D-F-F paths from (0, 0) to (6, k) where k = 0,..., c for c =. The number of paths from (0, 0) whch end at (6, 0), (6, 1) or (6, ) are ndcated n the dagram by the counts appearng next to the ponts. We observe that = paths end at (6, 0) ; = 1 paths end at (6, 1) ; end at (6, ). Next, we present our bjectve map between U-D paths and U-D-F-F paths. Fgure 4 S 3 wth connectng steps + = 1 paths

13 Transent results for M/M/1/c queues va path countng U-D paths to U-D-F-F paths In ths secton, we demonstrate one drecton of the bjecton. To move from U-D to U-D-F-F, we use a left sded algorthm. We llustrate by usng an example of an U-D path D D U U U U U D wth c = (Fgure 5a). 1 Consder the frst tme (from the left) that an U-D path crosses the horzontal axs ( y = 0 ) or level c (.e. y = c ). Replace that step by a flat component and reverse all subsequent steps (.e. U D). For our example, we obtan F U D D D D D U (Fgure 5b). For the new path, fnd the frst step that t crosses the horzontal axs or level c. Replace that frst step by a flat component. Reverse all subsequent steps. For our example, we obtan F U D F U U U D (Fgure 5c). 3 Contnue untl there are no more possble changes. In our example, we move to F U D F U U F U and then to F U D F U U F F (Fgure 5d). Fgure 5 Convertng an U-D path nto U-D-F-F

14 14 M. Hlynka, L.M. Hurajt and M. Cylwa 5. U-D-F-F paths to U-D paths In ths secton, we demonstrate the other drecton of the bjecton. To move from U-D-F-F to U-D, we use a rght sded algorthm. We llustrate wth an example of an U-D-F-F path F U D F U U F F wth c = (Fgure 5d). 1 If there s no F, do nothng to the path. Otherwse, at the F farthest to the rght, f the path s at level c (respectvely 0) at that pont, change the F to U (respectvely D). Smultaneously, any components farther to the rght get reversed (U D). In our example, we obtan F U D F U U F U (Fgure 5c). For the new path, fnd the F whch s now farthest to the rght. At ths F, f the path s at level c (respectvely 0) at that pont, change the F to U (respectvely D). Smultaneously, any components farther to the rght get reversed ( U D). In our example, we obtan F U D F U U U D (Fgure 5b). 3 Contnue untl there are no more possble changes. In our example, we move to F U D D D D D U U and then to D D U U U U U D (Fgure 5a). Thus, we have a bjecton between U-D and U-D-F-F paths. Moreover, ths bjecton has a specal feature that all U-D paths whch end at a partcular lattce pont wll map to U-D-F-F paths that end at a sngle lattce pont. The U-D paths are mapped to U-D-F-F paths wth a fnal heght that s obtaned by foldng over the level c + 1/ for heghts greater than c and foldng over 1/ for heghts less than 0. For example, f c = and k = 6, each path of U-D-F-F type wth sx steps wll end at heght 0 or 1 or. For k = 6, paths of U-D type wll end at heghts 6, 4,, 0,, 4, There are 6,,,,,, U-D paths endng at levels 6, 4,, 0,, 4, , respectvely. After applyng the bjecton, the correspondng U-D-F-F paths end at 0, 1,, 0, 1,, 0, respectvely. Thus, the number of U-D-F-F paths endng at, 1, 0, 6 6 respectvely whch are obtaned by summng, are +, 6 6 +, Countng U-D-F-F paths Our bjecton, gven n the prevous sectons allows us to count U-D-F-F paths by transformng them to U-D paths whch are easy to count. The general result to count U-D-F-F paths requres us to consder separate cases dependng on the value of k (even or odd) on the last step. The general result s as follows.

15 Transent results for M/M/1/c queues va path countng 15 Theorem 7. Consder all U-D-F-F paths of length k begnnng at level 0 and endng at level, ( = 0,, c) for gven c. The number of such paths s M 0, k f k + 0 mod j k+ mod ( c 1) j + ( k) =. k f k 1 mod j k++ 1 mod ( c+ 1) j + We apply multsectonng usng prmtve roots of unty to sum the seres n the prevous result (Rordan, 1968). The followng result appears n Guchard (1995). The greatest nteger functon appears n the upper lmt of the summaton. m 1 n+ 1 n n πk nπk πrk = + cos cos j. m m m m m j r mod m k= 1 n We apply ths result to Theorem 7 to get the followng corollary. Corollary 1. Consder all U-D-F-F paths of length k begnnng at level 0 and endng at level, ( = 0,, c) for gven c. Usng the greatest nteger functon [ ], we obtan the number of U-D-F-F paths from (0, 0) to ( k, ) wth upper heght lmt c s M 0, k c k k+ 1 π j k π j kπ j ( k) = + cos cos. c+ 1 c+ 1 c+ 1 c+ 1 c+ 1 j= 1 Although ths result does not seem to be a large mprovement, t actually s. Regardless of the sze of k, the number of summands n the expresson of Corollary 1 stays fxed. For small c, ths s a small sum. By usng the expresson n Corollary 1 together wth the expresson n Theorem 6, we have a convenent computatonal method for computng the transent probabltes. Of course, n practce, we would truncate the seres when the probabltes become suffcently small. 5.4 Fndng transent probabltes for M/M/1/c queues Example. Suppose, we have a smlar stuaton as n Example 1, now wth c =. Fnd the probablty of havng 0, 1 and customers after 6 mn (0.1 hours). Soluton. Snce λ = μ = 0 customers / hour, we can use Equaton (7) from Theorem 6. Usng a computer algebra system and truncatng at k = 0, we fnd the probabltes (to fve decmal places) to be: p00, (0. 1) = p01, (0. 1) = p (0. 1) = ,

16 16 M. Hlynka, L.M. Hurajt and M. Cylwa 6 Conclusons We have presented expressons for the transent probabltes for an M / M / 1/ c queueng system. For the case when λ = μ, the expressons allow for consderable smplfcaton. The expressons n ths case appear to be slghtly dfferent n appearance from, but equvalent to, the usual Bessel functon versons. Further, the smplcty of the analyss presented here makes the result very accessble. The λ = μ case s an mportant case. It gves good approxmatons for systems for whch λ s close to μ. Snce only transent probabltes are consdered, stablty condtons are not needed. Moreover, the nterest n heavy traffc models ( λ near μ wth λ < μ ) makes our study more useful. The specal contrbutons of ths work are the geometrc bjectons, presented n Sectons 4 and 5 that smplfy our path countng. These specal types of reflecton have specal nterestng characterstcs that ad consderably n our understandng of the process. In Secton 5, the counts of the number of paths endng n a partcular end pont surprsngly turns out to be the sum of every kth bnomal coeffcent. Ths allows us to use multsectonng and connects several dfferent concepts and results together. Although ths work s currently restrcted to the case λ = μ, the bjectons/ transformatons gven n Sectons 4 and 5 may have useful extensons far beyond ths case. In addton, the geometrc nterpretaton of (repeated) foldng over barrers suggests other possbltes for research, such as barrers that depend on a changng buffer sze. Some of the restrctons of the model could be further examned to see what other assumptons can be relaxed. 7 Acknowledgements The authors wsh to express ther grattude to the referee for the valuable suggestons whch helped to mprove the artcle. Ths research was partally funded through research grants of all authors from Natural Scences and Engneerng Research Councl (NSERC) of Canada. References Bohm, W., Krnk, A. and Mohanty, S.G. (1997) The combnatorcs of brth-death processes and applcatons to queues, Queueng System, Theory and Applcatons, Vol. 6, pp Champernowne, D.G. (1956) An elementary method of soluton of the queueng problem wth a sngle server and contnuous parameter, Journal of the Royal Statstcal Socety. Seres B (Methodologcal), Vol. 18, pp Grffths, J.D., Leonenko, G.M. and Wllams, J.E. (006) The transent soluton to M/E k /1 queue, Operatons Research Letters, Vol. 34, pp Gross, D., Shortle, J.F., Thompson, J.M. and Harrs, C.M. (008) Fundamentals of Queueng Theory (4th ed.). Hoboken, NJ: Wley. Guchard, D.R. (1995) Sums of selected bnomal coeffcents, College Mathematcs Journal, Vol. 6, pp Jan, J.L. and Mete, A.J. (005) Equvalence of dfferent expressons on transent solutons of M/M/1 queueng system, Int. J. Mathematcal Scences, Vol. 4, pp

17 Transent results for M/M/1/c queues va path countng 17 Jan, J.L., Mohanty, S.G. and Bohm, W. (007) A Course on Queueng Models. Boca Raton, FL: Chapman and Hall/CRC. Joy, M. and Jones, S. (005) Transent probabltes for queues wth applcatons to hosptal watng lst management, Health Care Management Scence, Vol. 8, pp Krnk, A. and Mortensen, C. (007) Transent probablty functons of fnte brth-death processes wth catastrophes, Journal of Statstcal Plannng and Inference, Vol. 137, pp Krshna Kumar, B. and Pava Madheswar, S. (005) Transent analyss of an M/M/1 queue subject to catastrophes and server falures, Stochastc Analyss and Applcatons, Vol. 3, pp Krshna Kumar, B. and Pava Madheswar, S. (007) Transent soluton of a catastrophc-cumrestoratve queueng problem wth correlated arrvals and varable servce capacty, Int. J. Informaton and Management Scences, Vol. 18, pp Margolus, B.H. (007) Transent and perodc soluton to the tme-nhomogeneous quas-brth process, Queueng Systems, Vol. 56, pp Rordan, J. (1968) Combnatoral Identtes. New York, NY: John Wley and Sons, p.131. Sharma, O.P. (1997) Markovan Queues. New Delh, Inda: Alled Publshers, p.189. Takacs, L. (1967) Combnatoral Methods n the Theory of Stochastc Processes. New York, NY: John Wley and Sons. Taraba, A.M.K. and El-Baz, A.H. (006) Exact transent solutons of nonempty Markovan queues, Computers and Mathematcs wth Applcatons. An Internatonal Journal, Vol. 5, pp Van Houdt, B. and Blonda, C. (005) Approxmated transent queue length and watng tme dstrbutons va steady state analyss, Stochastc Models, Vol. 1, pp