OPERATIONAL BEHAVIOR OF THE MAP/G/1 QUEUE UNDER N-POLICY WITH A SINGLE VACATION AND SET-UP 1

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1 Journal of Applied Mathematics and Stochastic Analysis, 15:2 (2002), OPRATIONAL BHAVIOR OF TH MAP/G/1 QUU UNDR NPOLICY WITH A SINGL VACATION AND STUP 1 HO WOO L Sung Kyun Kwan University Department of Systems Management ngineering Su Won, Korea mail: hwlee@yurimskkuackr BOO YONG AHN SK C&C Seoul, Korea (Received April, 2000; Revised June, 2001) This paper considers the MAP/G/1 queue under Rpolicy with a single vacation and setup We derive the vector generating functions of the queue length at an arbitrary time and at departures in decomposed forms We also derive the LaplaceStieltjes transform of the waiting time Computation algorithms for mean performance measures are provided Key words: MAP/G/1, Queue Length, Decomposition AMS subject classifications: 60K25, 90B22, 6M20 1 Introduction We consider the MAP/G/1 queue under Rpolicy with a single vacation and setup As soon as there are no customers to serve, the server takes a vacation (vacation period) After the vacation, if the server finds R or more customers, it starts a setup of a random length (setup period) If not, the server stays in the system until R customers accumulate (dormant period) After the setup, the server continuously serves the customers until the system empties (buy period) Thus, a cycle consists of a vacation period, a dormant period, a setup period and a busy period The service, vacation and setup times are assumed to be mutually independent with general distributions We also assume that they are independent of the phase of the underlying Markov chain Customers arrive according to the Markovian Arrival Process (MAP) with parameter matrices G and H For the formal definition of MAP, readers are advised to see Lucantoni, et al [10] We derive the vector generating functions of the queue length at an arbitrary time and at departures in decomposed forms We also derive the LaplaceStieltjes transform of the waiting time Computation algorithms for mean performance measures are provided 1 This work was supported in part by KOSF through Statistical Research Center for Complex Systems at Seoul National University Printed in the USA 2002 by North Atlantic Science Publishing Company 151

2 152 HO WOO L and BOO YONG AHN Lee et al [5] studied the MAP/G/1 system under multiple and single vacations with R policy and showed that the vector generating functions of the queue length at an arbitrary time and at departures are decomposed into two parts: one is the vector generating function of the queue length at an arbitrary idle time, and the other is an unidentified matrix generating function The authors [6] also confirmed that those decompositions hold for the MAP/G/1 system under Rpolicy with multiple vacations and setup We show that those decompositions also hold for this system with a single vacation and setup Kasahara et al [2] studied the MAP/G/1 queue with Rpolicy and multiple vacations They used the matrixanalytic approach pioneered by Neuts [11, 12], and his colleagues [ 10, 15, 16] In this paper, we employ the method of supplementary variables This method directly provides us with the queue length distributions at an arbitrary time Departure point queue length and virtual waiting time distributions can be obtained as byproducts For works in this category, see [1, 6, 1, 14] The remainder of the paper is organized as follows In Section 2, we develop the vector system equations and solve the equations by using the eigenvalues and eigenvectors of ÐG DH Ñ We derive the vector generating functions of the queue length at an arbitrary time and at departures in decomposed forms Interpretations of the decompositions are presented In Section, we derive the LaplaceStieltjes transform of the waiting time of an arbitrary customer In Section 4, we present the computation algorithms for mean performance measures We consider some special cases in Section 5 2 The Queue Length We first develop the system equations Let us define the following notations: RÀ steadystate queue length (number of customers in the system, including the one in service) NÀ the phase of the underlying Markov chain (UMC) WVßZVßLV À remaining service time, vacation time and setup time GßHÀ Ð Ñ parameter matrices of the Markovian arrival process G4ßH4À Ðß4Ñelement of GßH 1 œt<òn œó, ПŸÑ 1 œð1 ß1# ßáß1Ñ œ 1H/ À mean arrival rate œ IÐWÑÀ traffic intensity Ú Ý, (server is on a vacationñ #ß (server is in a dormant period) 'Ð>Ñ œ Û Ý $ß (server is in a setup) Ü %ß (server is busy) Let us define the following probabilities: ÐZ Ñ ß : ÐBÑœT<Ò' œßrœßn œßz ÐBßBÓÓß Ð!ߟŸÑ ÐWÑ ß : ß œ T <Ò' œ #ß R œ ß N œ Óß Ð! Ÿ Ÿ R ß Ÿ Ÿ Ñ : ÐBÑœT<Ò' œ$ßrœßn œßz ÐBßBÓÓß Ð R, ŸŸÑ ÐFÑ ß V : ÐBÑœT<Ò' œ%ßrœßn œßw ÐBßBÓÓ, РߟŸÑ V V

3 Operational Behavior of the MAP/G/ Queue 15 Then, defining and 2ÐBÑ as the pdf of the service time, vacation time and setup time respectively, the above probabilities satisfy the following equations: ÐZ Ñ!ß4 4!ß 4œ!œ : G : Ð!Ñ, ПŸÑ!œ : ÐZ Ñ G : H : Ð!Ñ, ПŸßŸŸÑ ß4 4 ß4 4 ß 4œ 4œ ÐFÑ : ÐBÑ œ ÐFÑ ÐFÑ : ÐBÑG : Ð!Ñ=ÐBÑ, Ð Ÿ Ÿ Ñ ß ß4 4 #ß 4œ ÐFÑ : ÐBÑ œ ÐFÑ ÐFÑ ÐFÑ : ÐBÑG : Ð!Ñ=ÐBÑ : ÐBÑH, 5ß 5ß4 4 5ß 5ß4 4 4œ 4œ Ð#Ÿ5ŸßŸŸÑ ÐFÑ : ÐBÑ œ ÐFÑ ÐFÑ ÐFÑ ÐWÑ : ÐBÑG : Ð!Ñ=ÐBÑ : ÐBÑH : Ð!Ñ=ÐBÑß Rß Rß4 4 ß ß4 4 Rß 4œ 4œ ПŸÑ ÐFÑ : ÐBÑ œ ÐFÑ ÐFÑ ÐFÑ ÐWÑ : ÐBÑG : Ð!Ñ=ÐBÑ : ÐBÑH : Ð!Ñ=ÐBÑß ß ß4 4 ß ß4 4 ß 4œ 4œ РߟŸÑ ÐZ Ñ : ÐBÑ œ ÐZ Ñ ÐFÑ : ÐBÑG : Ð!Ñ@ÐBÑ, Ð Ÿ Ÿ Ñ!ß 4œ!ß4 4 ß ÐZ Ñ : ÐBÑ œ ÐZ Ñ ÐZ Ñ : ÐBÑG : ÐBÑH, Ð ß Ÿ Ÿ Ñ ß ß4 4 ß4 4 4œ 4œ ÐWÑ : ÐBÑ œ ÐWÑ ÐZ Ñ : ÐBÑG : H 2ÐBÑ : Ð!Ñ2ÐBÑß Ð Ÿ Ÿ Ñ Rß Rß4 4 ß4 4 R 4œ 4œ ÐWÑ : ÐBÑ œ ÐWÑ ÐZ Ñ : ÐBÑG : H : Ð!Ñ2ÐBÑ, Ð R ß Ÿ Ÿ Ñ ß ß4 4 ß4 4 4œ 4œ Let us define the Ð Ñ row vectors as follows: ÐFÑ ÐFÑ ÐFÑ ÐFÑ ß ß# ß ß ß# ß : œ Ð: ß : ß á ß : Ñ, : ÐBÑ œ Ð: ÐBÑß : ÐBÑß á ß : ÐBÑÑ ÐZ Ñ ÐZ Ñ ÐZ Ñ ÐZ Ñ ÐWÑ ÐWÑ ÐWÑ ÐWÑ ß ß# ß ß ß# ß : ÐBÑ œ Ð: ÐBÑß : ÐBÑß á ß : ÐBÑÑ, : ÐBÑ œ Ð: ÐBÑß : ÐBÑß á ß : ÐBÑÑÞ Then, the above system equations can be converted to the vector system equations as follows: ÐZ Ñ!!! œ : G : Ð!Ñ Ð#ÞÑ

4 154 HO WOO L and BOO YONG AHN ÐZ Ñ,! œ : G : H : Ð!Ñ Ð Ÿ Ÿ R Ñ Ð#Þ#Ñ ÐFÑ ÐFÑ ÐFÑ # : ÐBÑ œ : ÐBÑG : Ð!Ñ=ÐBÑ Ð#Þ$Ñ ÐFÑ ÐFÑ ÐFÑ ÐFÑ : ÐBÑ œ : ÐBÑG : Ð!Ñ=ÐBÑ : ÐBÑH ß Ð# Ÿ Ÿ R Ñ Ð#Þ%Ñ ÐFÑ ÐFÑ ÐFÑ ÐFÑ ÐWÑ R R R : ÐBÑ œ : ÐBÑG : Ð!Ñ=ÐBÑ : ÐBÑH : Ð!Ñ=ÐBÑ Ð#Þ&Ñ ÐFÑ ÐFÑ ÐFÑ ÐFÑ ÐWÑ : ÐBÑ œ : ÐBÑG : Ð!Ñ=ÐBÑ : ÐBÑH : Ð!Ñ=ÐBÑ, Ð R Ñ Ð#Þ'Ñ ÐZ Ñ ÐZ Ñ ÐFÑ!! : ÐBÑ œ : ÐBÑG : Ð!Ñ@ÐBÑ Ð#Þ(Ñ ÐZ Ñ ÐZ Ñ ÐZ Ñ : ÐBÑ œ : ÐBÑG : H, Ð Ñ Ð#Þ)Ñ ÐWÑ ÐWÑ ÐZ Ñ R R R : ÐBÑ œ : ÐBÑG : H 2ÐBÑ : Ð!Ñ2ÐBÑ Ð#Þ*Ñ ÐWÑ ÐWÑ ÐWÑ ÐZ Ñ : ÐBÑ œ : ÐBÑG : ÐBÑH : Ð!Ñ2ÐBÑß Ð R ÑÞ Ð#Þ!Ñ Let us define the following vector generating functions ÐFÑ ÐFÑ ÐZ Ñ ÐZ Ñ : ÐDÑ œ : D, : ÐDß BÑ œ : ÐBÑD, : ÐDß BÑ œ : ÐBÑD ß œ ÐWÑ ÐWÑ 5 5 œr : ÐDß BÑ œ : ÐBÑD Þ Multiplying (2)(26) by D and summing over, yields ÐFÑ ÐFÑ : ÐDßBÑ œ : ÐDßBÑÐG DHÑ D ÐFÑ ÐFÑ ÐWÑ : ÐDß!Ñ : Ð!Ñ : ÐDß!Ñ =ÐBÑ Ð#ÞÑ Similarly, from (2) and (2), we get From (29) and (210), we get ÐZ Ñ ÐZ Ñ ÐFÑ : ÐDß BÑ œ : ÐDß BÑÐG DH Ñ : Ð!Ñ@ÐBÑÞ Ð#Þ#Ñ ÐWÑ ÐWÑ : ÐDßBÑ œ : ÐDßBÑÐG DHÑ Ð#Þ$Ñ R D ÐZ Ñ : H : Ð!ÑD 2ÐBÑÞ Let us define the Laplace transform as follows: œr

5 Operational Behavior of the MAP/G/ Queue 155 ÐFÑ B ÐFÑ ÐZ Ñ B ÐZ Ñ : ÐDß ) Ñ œ ' ) / : ÐDß BÑ : ÐDß ) Ñ œ ' ), / : ÐDß BÑß!! ÐWÑ B ÐWÑ : ÐDß ) Ñ œ ' ) / : ÐDß BÑ Taking the Laplace transforms of both sides of (211)(21), we obtain! ÐFÑ WÐÑ D : ÐDß ) ÑÐ) M G DH Ñ œ : ÐDß!Ñ ) ÐFÑ ÐWÑ ÐFÑ ) Ò: ÐDß!Ñ : Ð!ÑÓW Ð Ñ Ð#Þ%Ñ ÐZ Ñ ÐZ Ñ ÐFÑ : ÐDß ) ÑÐ) M G DH Ñ œ : ÐDß!Ñ : Ð!ÑZ Ð) Ñ Ð#Þ&Ñ ÐWÑ ÐWÑ : ÐDß ) ÑÐ) M G DH Ñ œ : ÐDß!Ñ œr ÐZ Ñ R : Ð!ÑD D : H L Ð) Ñ Ð#Þ'Ñ From (21) and (22), we get R ÐZ Ñ D : H œ : Ð!ÑD : ÐDÑÐG DHÑÞ Ð#Þ(Ñ Using (21) in (216), we get ÐWÑ ÐWÑ : ÐDß ) ÑÐ) M G DH Ñ œ : ÐDß!Ñ ÐZ Ñ Ò: ÐDÑÐG DH Ñ : ÐDß!ÑÓL Ð) ÑÞ Ð#Þ)Ñ ÐFÑ ÐZ Ñ ÐWÑ Now, we obtain : ÐDß!Ñ, : ÐDß!Ñ and : ÐDß!Ñ contained in (214), (215) and (21) They concern the queue length and the phase of the underlying Markov chain (UMC) at the ending points of a busy period, a vacation, and a setup This can be accomplished by eliminating the lefthand side of (214), (215) and (21) For this purpose, we use the eigenvalues of the matrix ÐG DHÑ and their right eigenvectors Let α ÐDÑ, Ðß#ßáßÑ be the eigenvalues Let 0ÐDÑ be the right eigenvector of αðdñ It is wellknown that α ÐDÑ and 0 ÐDÑ are related by (see Strang [1], for example) ÐG DHÑ0ÐDÑœαÐDÑ0ÐDÑ or Òα ÐDÑM G DH Ó0 ÐDÑœ!, ПŸÑ Using α ÐDÑ in ) of (215) and postmultiplying both sides by 0 ÐDÑ, we get ÐZ Ñ : ÐDß α ÐDÑÑÒα ÐDÑM G DHÓ0 ÐDÑ ÐZ Ñ ÐFÑ œ Ò: ÐDß!Ñ : Ð!ÑZ Ðα ÐDÑÑÓ0 ÐDÑÞ Ð#Þ*+Ñ Then, the lefthand side of (219 +Ñ vanishes and we get

6 156 HO WOO L and BOO YONG AHN ÐZ Ñ ÐFÑ : ÐDß!Ñ0 ÐDÑ œ : Ð!ÑZ Ð+ ÐDÑÑ0 ÐDÑÞ Ð#Þ*,Ñ quation Ð#Þ*,Ñ should hold for all eigenvalues α ÐDÑßáß α ÐDÑ Thus, we get : ÐZ Ñ ÐDß!ÑÒ0 ÐDÑß á ß 0ÐDÑ œ Ð!ÑÒZ Ðα ÐDÑÑ0 ÐDÑßáßZ Ðα ÐDÑÑ0 ÐDÑÓ : ÐFÑ Ð#Þ*Ñ Let us define an Ð Ñ matrix BÐDÑ of the eigenvectors as follows: BÐDÑ œ Ò0 ÐDÑß 0# ÐDÑßáß 0ÐDÑÓÞ It is proven in Lee et al [4] that the eigenvalues α ÐDÑß α# ÐDÑß á ß αðdñ are distinct for a D on ÖDÀ ± D ± Ÿ, and the inverse matrix B ÐDÑ exists Thus, (21Ñ becomes where ÐZ Ñ ÐFÑ 1 : ÐDß!Ñ œ : Ð!ÑBÐDÑH ÖZ Ðα ÐDÑÑ B ÐDÑ Ð#Þ*Ñ Î+! á! Ñ! + # á! Ö+ œ Ð ÓÞ ã ã ä ã Ï!! á + Ò H 1 Using Ð#Þ*Ñ in (215) yields ÐZ Ñ ÐFÑ 1 : ÐDß ) ÑÐ) M G DH Ñ œ : Ð!ÑBÐDÑH ÖZ Ðα ÐDÑÑ Z Ð) Ñ B ÐDÑ Ð#Þ#!Ñ In the same way, we get, from (214) and (21), ÐFÑ ÐWÑ ÐFÑ DW Ðα ÐDÑÑ 1 DW ÐαÐDÑÑ : ÐDß!Ñ œ Ò: ÐDß!Ñ : Ð!ÑÓBÐDÑH š B ÐDÑ Ð#Þ#Ñ ÐWÑ ÐZ Ñ 1 : ÐDß!Ñ œ Ò: ÐDÑÐG DH Ñ : ÐDß!ÑÓBÐDÑH eöl + ÐDÑÑf B ÐDÑ Ð#Þ##Ñ Using (221) in (214) and (222) in (21), we get ÐWÑ ÐZ Ñ : ÐDß) ÑÐ) M G DH Ñ œ Ò: ÐDÑÐG DH Ñ : ÐDß!ÑÓ H 1 BÐDÑ ÖL Ðα ÐDÑÑ L Ð) Ñ B ÐDÑ ÐFÑ ÐWÑ ÐFÑ : ÐDß ) ÑÐ) M G DH Ñ œ Ò: ÐDß!Ñ : Ð!ÑÓ Ð#Þ#$Ñ Ð#Þ#%Ñ DÒW Ðα ÐDÑÑW Ð) ÑÓ H 1 DW Ðα ÐDÑÑ BÐDÑ š B ÐDÑÞ quations (220), (22) and (224) will play central roles in obtaining the vector generating functions of the queue length and the LaplaceStieltjes transform (LST) of the waiting time

7 Operational Behavior of the MAP/G/ Queue 15 All the above quantities are expressed in eigenvalues, eigenvectors and diagonal matrices of the functions of the eigenvalues These quantities can be converted into familiar matrixanalytic notations See Appendix I for the conversions 21 The Queue Length at an Arbitrary Time The vector generating function ] ÐDÑ of the queue length at an arbitrary time can be obtained from ÐFÑ ÐZ Ñ ÐWÑ ] ÐDÑ œ : ÐDÑ : ÐDß ) Ñ : ÐDß ) Ñ : ÐDß ) Ñ Þ Ð#Þ#&Ñ ) œ! ) œ! ) œ! Using (220), (22) and (224) in (225), we get, after using ( M2) of Appendix I, ] ÐDÑ œ : ÐDÑBÐDÑH ÖL Ðα ÐDÑÑ B ÐDÑ ÐFÑ 1 1 Z Ðα ÐDÑÑL Ðα ÐDÑÑ α ÐDÑ : Ð!ÑBÐDÑH š B ÐDÑ Ð#Þ#'Ñ ÐDÑW Ðα ÐDÑÑ H 1 W Ðα ÐDÑÑDÑ BÐDÑ š B ÐDÑÞ The number of customers at an arbitrary time during a dormant period can be obtained from the information concerning the previous busy period termination point and the number of customers that arrive during the vacation Thus, we can express : ÐDÑ in (226) in terms ÐFÑ of : Ð!Ñ Let ÐZÑ4be the probability that customers arrive during a vacation and the phase of the UMC is 4 at the end of the vacation, given that the UMC is at the start of the vacation Let Z be the matrix of ÐZÑ4 and Z ÐDÑ be the matrix generating function of Z such that Z ÐDÑ œ ZD Then, we have the following theorem Theorem 1 : If we define the matrix as F F 5 œ Z, F œ Z ÒÐ GÑ HÓ ß Ð Ñß Ð#Þ#(Ñ!! 5 5œ! Ð+Ñ we get ÐFÑ : ÐDÑ œ : Ð!Ñ F Ð GÑ D Ð#Þ#)Ñ Ð,Ñ ÐF Ñ 4 is the probability that the system state Ðqueue length Ñ visits during the dormant period and the phase of the UMC at the visit is 4, given that the vacation starts with the phase in ÐZ Ñ Proof: Ð+Ñ For œ!, we get, from (21), :! G œ :! Ð!ÑÞ The queue length at the end of the vacation is the number of customers that arrive during the vacation Thus, ÐZ Ñ ÐFÑ ÐZ Ñ ÐFÑ we get : ÐDß!Ñ œ : Ð!ÑZ ÐDÑ which means : Ð!Ñ œ : Ð!ÑZ With œ!, we get ÐZ Ñ ÐFÑ ÐFÑ ÐFÑ :! Ð!Ñ œ : Ð!ÑZ!, which implies :! G œ : Ð!ÑZ! œ : Ð!ÑF! ÐFÑ from the definition Thus we get : œ : Ð!ÑF ÐGÑ For, we get, from (22),!! ÐFÑ : G œ : Ð!Ñ 5 ÐFÑ Z ÒÐGÑ H Ó œ : Ð!ÑF Þ 5 5œ! ÐFÑ Thus, we get : œ : Ð!ÑF ÐGÑ, which implies (22)

8 15 HO WOO L and BOO YONG AHN Ð,Ñ We first notice that Ð GÑ H is the phase change probability during an interarrival time Conditioning on the number of customers that arrive during the vacation completes the proof Using (22) in (226), ] ÐDÑ becomes ÐFÑ ] ÐDÑ œ : Ð!Ñ FÐ GÑ D BÐDÑH1ÖL ÐαÐDÑÑ B ÐDÑ Z Ðα ÐDÑÑL Ðα ÐDÑÑ H 1 α ÐDÑ BÐDÑ š B ÐDÑ Ð#Þ#*Ñ ÐDÑW Ðα ÐDÑÑ H 1 W Ðα ÐDÑÑDÑ BÐDÑ š B ÐDÑÞ ÐFÑ Now, we consider the term : Ð!Ñ contained in (229) We note that : Ð!Ñ is the rate at which the system becomes empty If we let B! be the vector probability that there are no customers in the system at an arbitrary departure, this should be equal to B! because, which is the arrival rate, should be equal to the departure rate in steady state Thus, we have Using (20) in (229), we get ÐFÑ! : Ð!Ñ œ B Þ Ð#Þ$!Ñ ] ÐDÑ œ B! FÐ GÑ D BÐDÑH1ÖL ÐαÐDÑÑ B ÐDÑ ÐFÑ Z Ðα ÐDÑÑL Ðα ÐDÑÑ H 1 α ÐDÑ BÐDÑ š B ÐDÑ Ð#Þ$+Ñ ÐDÑW Ðα ÐDÑÑ H 1 W Ðα ÐDÑÑDÑ BÐDÑ š B ÐDÑÞ Now, using Appendix I, we can convert (21 +Ñ into the matrixanalytic notations as follows: ] ÐDÑœ ÐDÑB! œ FÐGÑ D LÐDÑÒZ ÐDÑLÐDÑMÓÐG DHÑ ÐDÑÒDM ÐDÑÓ Ð#Þ$,Ñ where ÐDÑ and LÐDÑ are the matrix generating functions of the number of customers that arrive during a service time and a setup time, respectively Now, we need to find B! Obtaining B! requires the distribution of the number of customers that are served between two neighboring busyperiod termination points (We will call this interval a cycle) This, in turn, requires the queue length distribution at the busy period initiation point

9 Operational Behavior of the MAP/G/ Queue 159 Theorem 2: Let U Rßsetup ÐDÑ be the matrix generating function of the queue length at the starting epoch of an arbitrary busy period, given the phase at the ending epoch of the previous busy period Then, we have U setupðdñ œ F Ð GÑ D ÐG DHÑLÐDÑ Z ÐDÑLÐDÑÞ Ð#Þ$#Ñ Rß Proof: Let U R ÐDÑ be the matrix generating function of the queue length at an arbitrary busy period starting epoch of the MAP/G/1 queue with Rpolicy and a single vacation (without setup) Conditioning on the number of customers that arrive during the vacation, we get From Theorem 1, we have R5 R U ÐDÑ œ Z ÐDÑ Z D Z ÒÐ GÑ HÓ D Þ Ð#Þ$$Ñ R 5 5œ! 5 F ÐGÑ D œ Z5ÒÐGÑ HÓ ÐGÑ D 5œ! 5 œ 5œ! œ Ð Ñ D 5 Z G Z ÒÐGÑ HÓ ÐG Ñ D Postmultiplying both sides of the above equation by ÐGÑ, we get Using (24) in (2), we have 5 œ 5œ! 5 Z D œ F D Z ÒÐ GÑ HÓ D Þ Ð#Þ$%Ñ 5 œ 5œ! URÐDÑ œ Z ÐDÑ œf D Z5ÒÐ GÑ HÓ D Z ÒÐGÑ HÓ D 5œ! 5 R5 R R 5 5 œ 5œ! œ Z ÐDÑF D Z ÒÐGÑ HÓ D œ Z ÐDÑ ÐGÑ D G 5 F œ Z5ÒÐGÑ HÓ ÐGÑ D HDÞ œ! 5œ! Using (22), we get U ÐDÑ œ F Ð GÑ D ÐG HDÑ Z ÐDÑÞ Ð#Þ$&Ñ R

10 160 HO WOO L and BOO YONG AHN Then, URßsetupÐDÑ œ URÐDÑLÐDÑ completes the proof Now, define OÐDÑ as the matrix generating function of the number of customers that are served during a cycle Then using (22), we have OÐDÑ œ U ÐDÑ ± Rßsetup DœKÐDÑ Ð#Þ'Ñ œ FÐ GÑ ÒKÐDÑÓ ÒG HKÐDÑÓ Z ÐKÐDÑÑ LÐKÐDÑÑ ' ÐGHKÐDÑÑB where Z ÐKÐDÑÑ œ! / ZÐBÑ (Lucantoni, et al [10]) and KÐDÑ is the matrix generating function of the number of customers that are served during a fundamental period (Neuts [12]) Let O œ OÐDÑ ± Dœ be the matrix of the phase change probability of the UMC during an arbitrary cycle Then, from (26) we get O œ OÐDÑ ± Dœ œ F Ð GÑ K ÐG HKÑ Z ÐKÑ LÐKÑ Ð#Þ$(Ñ Let, be the stationary vector of O such that, œ, O and,/ œ Let, œ D O ÐDÑ ± Dœ / be the mean number of customers that are served during a cycle, given the phase at the ending epoch of a busy period Then, it is well known that B! can be obtained from (Neuts [12]), B! œ,, Þ Ð#Þ$)Ñ Now, we can express B! in terms of F defined and interpreted in Theorem 1 We first obtain, Theorem : We have, œ FÐGÑ IÐZÑM IÐLÑM / Ð#Þ$*Ñ ÐO M ÑÐ/1G HKÑ H Þ Proof: See Appendix III Using (29) in (2) with,ðo M Ñ œ! yields B! ÐÑ,, FÐG Ñ / IÐLÑ œ Þ Ð#Þ%!Ñ Using (240) in (21 +ß,Ñ, we get ] ÐDÑ œ < ÐDÑ BÐDÑH š B ÐDÑ 1 ÐÑÐDÑW Ðα ÐDÑÑ W Ðα ÐDÑÑD Ð#Þ%Ñ œ < ÐDÑÐ ÑÐD ÑÐDÑÒDM ÐDÑÓ #

11 Operational Behavior of the MAP/G/ Queue 161 where < ÐDÑ œ, F ÐG Ñ / IÐLÑ, F Ð GÑ D, BÐDÑH1š B ÐDÑ Ð#Þ%#Ñ Z Ðα ÐDÑÑ αðdñ œ L ÐαÐDÑÑ IÐLÑ, FD ÐG HDÑ Z ÐDÑBÐDÑH1š B ÐDÑ IÐLÑα ÐDÑ and < # ÐDÑ œ, F ÐG Ñ / IÐLÑ ÒZ ÐDÑMÓÐGDHÑ, F ÐGÑ D IÐZÑ, Ð#Þ% Ñ IÐLÑ, œ FD ÒLÐDÑMÓÐGDHÑ ÐG HDÑ Z ÐDÑ We note that < ÐDÑ is just the matrixanalytic expression of < ÐDÑ and they are equivalent # 22 The Queue Length at Departures All the information concerning the queue length at departures can be recover from : ÐFÑ ÐDß!Ñ, because the departure epochs are the epochs at which the remaining service times become zero Thus, after a normalization and discounting the departing customer, we get the vector generating function \ÐDÑ of the queue length at departures as ÐFÑ ÐFÑ : ÐDß!Ñ : ÐDß!Ñ : ÐFÑÐß!Ñ/ IÐLÑ \ÐDÑ œ D œ D Ð#Þ%%Ñ ÐFÑ ÐFÑ ÐFÑ where we used œ : Ðß!Ñ/ œ : ÐDß!Ñ ± Dœ /, because : Ðß!Ñ/ œ ÐFÑ œ : Ð!Ñ/ is the departure rate of the customers which is equal to the arrival rate in steady state After using (219Ñ, (222), (22) and (20) in (221), we can rewrite : ÐFÑ ÐDß!Ñin a different form as ÐFÑ : ÐDß!Ñ œ B! FÐ GÑ D BÐDÑH1ÖL ÐαÐDÑÑ B ÐDÑ Z Ðα ÐDÑÑL Ðα ÐDÑÑ H 1 α ÐDÑ BÐDÑ š B ÐDÑ Ð#Þ%&Ñ Dα ÐDÑW Ðα ÐDÑÑ H 1 W Ðα ÐDÑÑD BÐDÑ š B ÐDÑÞ Using (245) in (244) together with (240), we get

12 162 HO WOO L and BOO YONG AHN \ ÐDÑ œ < ÐDÑ BÐDÑH š B ÐDÑ 1 Ðα Ñ ÐDÑW Ðα ÐDÑÑ ÒW Ðα ÐDÑÑDÓ Ð#Þ%'Ñ œ < ÐDÑ ÐG DHÑÐDÑÒDM ÐDÑÓ # ÐÑ where < ÐDÑ and < # ÐDÑ were given in (242) and (24) From (241) and (246), we confirm the well known relationship (Takine and Takahashi [1]), 2 Decompositions and Probabilistic Interpretations ] ÐDÑÐG DH Ñ œ ÐD Ñ\ ÐDÑÞ Ð#Þ%(Ñ To interpret < ÐDÑ (or equivalently, < # ÐDÑÑ, we need the mean length of the idle period MRß setup which consists of a vacation, a dormant period and a setup time Let X ÐÑ ) be the matrix LST of the time until customers arrive given the phase of the UMC at time 0 Then, we easily get! X ÐќР) ) M GÑ HX ÐÑÐ ), ÑÐ, X ÐÑœ ) MÑ Ð#Þ%)Ñ R where Ð) M GÑ H is the LST of the time length until the first arrival Let M Ð) Ñ be the matrix Laplace transform of the idle period of the MAP/G/1 queue under Rpolicy with a single vacation (without setup), given the phase at the end of a busy period Then, conditioning on the number of arrivals during the vacation, we get M ÐÑœ ) Z ÐÑ ) X ÐÑ ) Z ÐÑ ) Ð#Þ%*Ñ R 5 R5 5 5œ! 5œR where 5 ÐÑ ) is the matrix Laplace transform of the vacation length which includes the probability that 5 customers arrive during the vacation Let ß be the mean time until customers arrive, given that the UMC is in phase at time X 0 Define a column vector œð ß ß ß# ßáß ß Ñ Then using (24), we can easily derive Z ÐÑ œ! 5 œ X Ð) Ñ ± ) / œ ÒÐ GÑ HÓ Ð G Ñ / Þ Ð#Þ&!Ñ 5œ! Let ( Rß be the mean length of the idle period in the MAP/G/1 queue under Rpolicy and a single vacation (without setup), given that the UMC is in phase at the end of a busy period X Let us define the column vector ( R œð( Rß ß> Rß# ßáß( Rß Ñ Then, we can get Then, from (249), we get ( œ M Ð) Ñ ± / œ F Ð G Ñ / / Þ Ð#Þ&Ñ R ) R ) œ! IÐM Ñœ ÐÑ œ R M ) /, F5ÐG Ñ / IÐZÑÞ Ð#Þ&#Ñ ) R ) œ! 5œ! Then, we get the mean length of an idle period as

13 Operational Behavior of the MAP/G/ Queue 16 If we let IÐF (2) and (240), IÐM setup Ñ œ IÐM Ñ IÐLÑ œ, F Ð G Ñ / IÐLÑÞ Ð#Þ&$Ñ Rß R Rßsetup Ñ be the mean length of an arbitrary busy period, we easily get, from IÐFRßsetup Ñ œ,, IÐWÑ œ, F5Ð G Ñ / IÐLÑ Þ Ð#Þ&%Ñ 5œ! Now, we are ready to interpret the queue length generating functions given in (241) and (246) If the system state during the dormant period visits, it stays there for ÐG Ñ / on the average Thus, FÐG Ñ / is the mean length of the dormant period during an arbitrary idle period Thus, without proofs, we have Theorem 4 and Theorem 5 ÐZ Ñ Theorem 4: Let 9 and 9 be the probabilities that the server is in a dormant period and in a setup period given that the system is idle Then we get 9, FÐG Ñ / œ, F ÐG Ñ / IÐLÑ 9 ÐZ Ñ œ, F ÐG Ñ / IÐLÑ Ð#Þ&&+Ñ Ð#Þ&&,Ñ 9 ÐWÑ IÐLÑ, F ÐG Ñ / IÐLÑ œ Þ Ð#Þ&&Ñ Theorem 5: Let < ß be the probability that at an arbitrary time during a dormant period, there are customers in the system and the phase of the UMC is in Define the vector œð< ß< ßáß< Ñ Then we get < ß ß# ß,FÐGÑ <, F ÐG Ñ / œ ß Ð!ŸŸÑÞ Ð#Þ&'Ñ 5œ! 5 Now, using (255 +ß,ß Ñ and (256), the vector generating function ] ÐDÑ of the queue length at an arbitrary time Ð(241) Ñ and \ ÐDÑ at departures Ð(246) Ñ become ] ÐDÑ œ œ ÐZ Ñ ÐWÑ D Z ÐDÑ 9 < 9, 9, FD ÐG DHÑ Z ÐDÑ L ÐDÑ ÐÑÐDÑW Ðα ÐDÑÑ H 1 W Ðα ÐDÑÑD BÐDÑ š B ÐDÑ Ð#Þ&(Ñ œ œ ÐZ Ñ ÐWÑ D ÐDÑ 9 < 9, Z 9, FD ÐG DHÑ Z ÐDÑ L ÐDÑ Ð ÑÐD ÑÐDÑÒDM ÐDÑÓ

14 164 HO WOO L and BOO YONG AHN \ ÐDÑœœ ÐZ Ñ ÐWÑ D Z ÐDÑ 9 < 9, 9, FD ÐG DHÑZ ÐDÑ L ÐDÑ Ðα Ñ ÐDÑW Ðα ÐDÑÑ H 1 ÒW Ðα ÐDÑÑDÓ BÐDÑ š B ÐDÑ Ð#Þ& Ñ œ œ ÐZ Ñ ÐWÑ D ÐDÑ 9 < 9, Z 9, FD ÐG DHÑ Z ÐDÑ L ÐDÑ ÐÑ ÐG DHÑÐDÑÒDM ÐDÑÓ where Z ÐDÑ and L ÐDÑ are the matrix generating functions of the number of customers that arrive during a vacation time and a setup time, respectively (see Appendix I) Now, we are ready to interpret the term œ ÐZ Ñ ÐWÑ < D, Z ÐDÑ, FD ÐG DHÑZ ÐDÑ L ÐDÑ Ð#Þ&*Ñ which is common to (25) and (25) In the following theorem, we show that (259) is the vector generating function of the queue length at an arbitrary idle time Theorem 6: Let : idle ÐDÑ be the vector generating function of the queue length at an arbitrary idle time Then we have : idle ÐDÑ œ Ð#Þ'!Ñ œ ÐZ Ñ ÐWÑ < D, Z ÐDÑ, FD ÐG DHÑZ ÐDÑ L ÐDÑ Proof: From Theorem 4, the system is in a dormant period with probability 9, in a ÐZ Ñ ÐWÑ vacation with probability 9, or in a setup period with probability 9 If the system is in a dormant period, there are customers in the system with probability < which accounts for the first term If the system is in a vacation at an arbitrary time, the queue length is the number of customers that arrive during the elapsed vacation time, which accounts for the second term If the system is in a setup period, the queue length is the number of customers at the setup period starting point (see quation (25) for this), plus the number of customers that arrive during the elapsed setup time, which accounts for the third term Applying (260) into (25) and (25), we get the final decomposed forms of the vector generating function of the queue length distributions as where ] ÐDÑ œ : idle ÐDÑ ; ] ÐDÑ Ð#Þ'Ñ \ ÐDÑ œ : idle ÐDÑ ; \ ÐDÑ Ð#Þ'#Ñ

15 Operational Behavior of the MAP/G/ Queue 165 ÐÑÐDÑW ÐαÐDÑÑ ; ] B 1 W Ðα ÐDÑÑD ÐDÑ œ ÐDÑH š B ÐDÑ Ð#Þ'$Ñ and œ Ð ÑÐD ÑÐDÑÒDM ÐDÑÓ Ðα Ñ ÐDÑW ÐαÐDÑÑ ; \ B 1 ÒW Ðα ÐDÑÑDÓ ÐDÑ œ ÐDÑH š B ÐDÑ ÐÑ œ ÐG DHÑÐDÑÒDM ÐDÑÓ Þ Ð#Þ'%Ñ Decomposition results of (261) and (262) confirm the results of Lee and Ahn [5, 6] The Waiting Time In this section, we derive the LST of the waiting time of an arbitrary arriving customer (actual waiting time) Let us define the following notations [ Ð ) ÑÀ the LST of the actual waiting time of an arbitrary arriving customer ÐFÑ [ Ð) ÑÀ the LST of the actual waiting time of the customer who arrives during the busy period (including the probability that the server is busy when the customer arrives) [ Ð) ÑÀ the LST of the actual waiting time of the customer who arrives during the dormant period (including the probability that the server is in a dormant period when the customer arrives) ÐZ Ñ [ Ð) ÑÀ the LST of the actual waiting time of the customer who arrives during the vacation (including the probability that the server is busy when the customer arrives) ÐFÑ ) First, we obtain [ Ð Ñ The virtual waiting time in this case is the sum of ÐÑ the remaining service and ÐÑ the service times of the waiting customers Information concerning ÐÑ and ÐÑ is contained in : ÐFÑ ÐDß ) Ñ in (224) Using (222), (20), (240) and Ð) M G DHÑ œ BÐDÑH1š ) α ÐDÑ B ÐDÑ, we can rewrite (224) as : ÐFÑ ÐDß ) Ñ œ Ð Ñ Thus, the LST ÐGÑ ÐDÑL Ð ÐDÑÑ 9 < D BÐDÑH1š α α B ÐDÑ ) α ÐDÑ 9, BÐDÑH š B ÐDÑ Ð$ÞÑ [ Ð Ñ ÐFÑ ) ÐZ Ñ Z Ðα ÐDÑÑL Ðα ÐDÑÑ 1 Ò) αðdñó DÒW H Ðα ÐDÑÑW Ð) ÑÓ 1 W Ðα ÐDÑÑD BÐDÑ š B ÐDÑ becomes

16 166 HO WOO L and BOO YONG AHN ÐFÑ ÐFÑ : ÐDß) Ñ H D DœW Ð) Ñ / [ Ð ) Ñ œ ¹ œ Ð Ñ 9 < ÒW Ð) ÑÓ B ÐW Ð) ÑÑH1š α ) α ) B ÐW Ð) ÑÑ Ð$Þ#Ñ ÐW Ð ÑÑL Ð ÐW Ð ÑÑÑ ) α ÐW Ð) ÑÑ ÐZ Ñ Z Ðα ÐW Ð) ÑÑÑL Ðα ÐW Ð) ÑÑÑ H 1 Ò) α ÐW Ð) ÑÑÓ 9, B ÐW Ð) ÑÑH š B ÐW Ð) ÑÑ / where HÎprobabilities is the factor that converts the virtual phase probabilities to actual phase Now, we obtain [ Ð) Ñ If the virtual customer arrives during the dormant period and sees 5 customers, his waiting time is the sum of ÐÑ the service times of those 5 customers, ÐÑ the time until ÐR5Ñmore customers arrive and ÐÑ the setup time The system is idle with probability Ð Ñ Thus Ð Ñ9 < D 5 5 is the joint vector generating function of the number of customers that are found by the virtual customer who arrives during the dormant period Using (24), we get 5 [ Ð) Ñ œ Ð Ñ9 H < 5ÒW Ð) ÑÓ X R5 Ð) Ñ /L Ð) Ñ Ð$Þ$Ñ ÐZ Ñ 5œ! Now, we obtain [ Ð) Ñ Consider the virtual customer who arrives during a vacation Suppose 5 customers arrive during the elapsed vacation and 6 customers arrive during the remaining vacation Then we have two cases: Case 1: If 56 R, the virtual waiting time is the sum of ÐÑ the remaining vacation time, ÐÑ the 5 service times and ÐÑ the setup time Case 2: If 56Ÿ, the virtual waiting time is the sum of ÐÑ the remaining vacation time, ÐÑ the time until RÐ56Ñmore customers arrive, ÐÑ the 5 service times and Ð@Ñ the setup time Obtaining the LST of the actual waiting time due to ÐÑ and ÐÑ of Case 1 and ÐÑ and ÐÑ of Case 2 requires the joint transform of the number of customers that arrive during the elapsed vacation time, the number of customers that arrive during the remaining vacation, and the length of the remaining vacation time For this purpose, we employ the approach used in Kasahara, et al [2], which is presented in Appendix II To compute the waiting time due to ÐÑ of Case 2, we can use (24) Then, the LST of the actual waiting time during the vacation can be expressed as ÐZ Ñ [ ÐќР) Ñ9, H5ß6ÐÑÒWÐÑÓ ) ) R# R5# 5œ! ÐZ Ñ 5 5œ! 6œmaxÐR5ß!Ñ 5 H ÐÑÒWÐÑÓ ) ) X ÐÑ ) /L ÐÑ ) Ð$Þ%Ñ 5œ! 6œ! 5ß6 R56

17 Operational Behavior of the MAP/G/ Queue 16 ÐZ Ñ 5 œð9 Ñ, œ H 5ß6 Ð) ÑÒW Ð) ÑÓ R# R5# 5œ! 6œ! 5 H Ð) ÑÒW Ð) ÑÓ ÒX Ð) ÑM Ó/L Ð) Ñ 5œ! 6œ! 5ß6 R56 To simplify the term 5 ÒW Ð) ÑÓ H Ð) Ñ/ contained in (4), let = ÐDß ) Ñ be 5œ! 6œ! 5ß6 the joint transform of the number of customers that arrive during the elapsed vacation (based on the actual arrival point) and the length of the remaining vacation time with the phase at the start of the vacation being in Let its column vector be = ÐDß) Ñ Then, from (II1) of Appendix II, we have = ÐDß) ÑœH ÐDßDß) Ñ / ± # # DœDßDœ Ð$Þ&+Ñ B œ ' ' ÐGDHÑ> H / ÐB>Ñ ) M ÐG H Ñ // >Þ Bœ! >œ! œ x From ÐG H Ñ / œ!, Ð Ñ and (I) of Appendix I, (5 +Ñis reduced to Then, we have = Z Ð ÐDÑÑZ Ð Ñ ) B 1š α ) H Ò) α ÐDÑÓ B ÐDß Ñ œ ÐDÑH ÐDÑ / Þ Ð$Þ&,Ñ 5œ! 6œ! 5 ÒWÐÑÓ ) H ÐÑ ) / œ= ÐWÐÑß ) ) Ñ 5ß6 Ð$Þ'Ñ Using (6) in (4), we get ÐZ Ñ Z Ðα ÐW Ð) ÑÑÑZ Ð) Ñ H 1 Ò) α ÐW Ð) ÑÑÓ œ B ÐW Ð) ÑÑH š B ÐW Ð) ÑÑ / Þ ÐZ Ñ Z Ðα ÐW Ð) ÑÑÑZ Ð) Ñ H 1 Ò) α ÐW Ð) ÑÑÓ [ Ð) Ñ œ Ð Ñ9, šb ÐW Ð) ÑÑH š B ÐW Ð) ÑÑ / R#R5# 5 H Ð) ÑÒW Ð) ÑÓ ÒX Ð) Ñ M Ó/ L Ð) ÑÞ 5œ! 6œ! 5ß6 R56 Ð$Þ(Ñ ÐWÑ ) Now, we obtain [ Ð Ñ The virtual waiting time in this case is the sum of ÐÑ the remaining setup time and ÐÑ the service times of the waiting customers We can use (22) for this purpose Using (219Ñ, Ð#Þ#)Ñ, (20), (2), and Ð) M G DHÑ œ BÐDÑH1š ) α ÐDÑ B ÐDÑ, quation (22) becomes

18 16 HO WOO L and BOO YONG AHN ÐWÑ α ÐDÑ ) αðdñ : ÐDß ) Ñ œ Ð Ñ 9 < D BÐDÑH1š B ÐDÑ ÐZ Ñ Z Ðα ÐDÑÑ 1 Ò) αðdñó 9, BÐDÑH š B ÐDÑ Ð$Þ)Ñ Thus, we get H 1 B ÐDÑ ÖL Ð) Ñ L Ðα ÐDÑÑ B ÐDÑÞ ÐWÑ ÐWÑ H DœW Ð) Ñ [ Ð) Ñ œ : ÐDß) Ñ ± / œ ÐÑ ÐW Ð ÑÑ 9 < ÒW Ð) ÑÓ B ÐW Ð) ÑÑH1š α ) B ÐW Ð) ÑÑ Ð$Þ*Ñ ) α ÐW Ð) ÑÑ ÐZ Ñ Z Ðα ÐW Ð) ÑÑÑ 1 Ò) α ÐW Ð) ÑÑÓ 9, B ÐW Ð) ÑÑH š B ÐW Ð) ÑÑ 1 B ÐW Ð) ÑÑH ÖL Ð) Ñ L Ðα ÐW Ð) ÑÑÑ B ÐW Ð) ÑÑ / Þ Then, using (2), (), () and (9), we get the LST of the actual waiting time as ÐZ Ñ ÐWÑ ÐFÑ [ Ð) Ñœ[ Ð) Ñ[ Ð) Ñ[ Ð) Ñ[ Ð) Ñ H œœ 5 ÐZ Ñ Z Ð) Ñ ÐWÑ L Ð) Ñ 9 < 5ÒWÐÑÓLÐÑ ) ) 9, LÐÑ ) 9, 5œ! ) IÐLÑ) H B ÐW Ð) ÑÑH š B ÐW Ð) ÑÑ / Ð$ 10+Ñ 1 Ð) Ñ ) α ÐW Ð) ÑÑ Ð9 Ñ 5 H < 5ÒW Ð) ÑÓ Ò X R5 Ð) Ñ M Ó / L Ð) Ñ 5œ! R# R5# ÐZ Ñ 5 Ð9 Ñ, ÒWÐÑÓ ) H 5ß6 ÐÑÒ ) X R56 ÐÑ ) M ÓLÐÑ / ) or, in matrixanalytic notations, 5œ! 6œ! 5 ÐZÑ Z Ð Ñ ÐWÑ L Ð Ñ [ÐÑœ ) œ9 < 5ÒWÐÑÓLÐÑ ) ) 9, ) LÐÑ ) 9, ) 5œ! ) IÐLÑ)

19 Operational Behavior of the MAP/G/ Queue 169 H Ð)) Ñ Ò M G W Ð) ÑH Ó / Ð$ 10,Ñ Ð9 Ñ 5 H < 5ÒW Ð) ÑÓ Ò X R5 Ð) Ñ M Ó / L Ð) Ñ 5œ! R# R5# ÐZ Ñ 5 Ð9 Ñ, ÒWÐÑÓ ) H 5ß6 ÐÑÒ ) X R56 ÐÑ ) M ÓLÐÑ / ) 5œ! 6œ! 4 Mean Performance Measures In this section, we derive the algorithm to compute the mean queue length and the mean waiting time 41 The Mean Queue Length ÐÑ We can get the mean queue length PH œ \ ÐÑ/ at departures by following the standard procedure presented in Lucantoni, et al [10] We first rewrite (246) as ÐZ Ñ \ ÐDÑÒDM ÐDÑÓ œ œ9 < D ÐG DHÑ 9, ÒZ ÐDÑ MÓ Ð%ÞÑ ÐWÑ 9, FÐ GÑ D ÐG DHÑ Z ÐDÑ ÒLÐDÑ MÓÐDÑÞ IÐLÑ Using Dœ in (41) and adding \ ÐÑ/1 to both sides, we get ÐZ Ñ \ ÐÑ œ 1 œ9 < ÐG HÑ 9, ÐZ MÑ ÐWÑ 9, FÐGÑ ÐG HÑZ ÐL MÑ Ð%Þ#Ñ IÐLÑ M Ð /1Ñ Þ Let Y ÐDÑ be the righthand side of (41) Then, we get ÐÑ Y ÐÑ œ œ9 < ÐG HÑ 9 < H Ð%ÞÑ ÐZ Ñ 9,Z ÐÑ ÐÑ

20 10 HO WOO L and BOO YONG AHN ÐWÑ ÐÑ 9, FÐGÑ ÐG HÑFÐGÑ HZ ÐÑ ÐL MÑ IÐLÑ ÐWÑ ÐÑ 9, FÐGÑ ÐG HÑZ L Ð+Ñ IÐLÑ ÐZ Ñ œ9 < ÐG HÑ 9, ÐZ MÑ and ÐWÑ ÐÑ 9, FÐGÑ ÐG HÑZ ÐL MÑ ÐÑ IÐLÑ Ð#Ñ ÐZ Ñ Ð#Ñ Y ÐÑ/ œ œ# 9 < H/ 9, Z ÐÑ/ Ð%Þ4Ñ 2 ÐWÑ ÐÑ ÐÑ 9, FÐGÑ ÐG HÑFÐGÑ HZ ÐÑ L ÐÑ/ IÐLÑ ÐWÑ ÐÑ 9, FÐGÑ ÐG HÑZ L ÐÑ/ IÐLÑ #ÐÑ œ9 < ÐG HÑ9 < H ÐZ Ñ 9,Z ÐÑ ÐÑ ÐWÑ ÐÑ 9, FÐGÑ ÐG HÑFÐGÑ HZ ÐÑ ÐL MÑ IÐLÑ ÐWÑ ÐÑ ÐÑ 9, FÐGÑ ÐG HÑZ L ÐÑ ÐÑ/ IÐLÑ

21 Operational Behavior of the MAP/G/ Queue 11 R ÐZ Ñ œ9 < ÐG HÑ 9, ÐZ MÑ ÐWÑ Ð#Ñ 9, FÐGÑ ÐG HÑZ ÐL MÑ ÐÑ/ Þ IÐLÑ Then, we get the mean queue length at departures as P œ \ ÐÑ/ H Ð#Ñ ÐÑ Ð#Ñ œ \ #ÐÑ ÐÑ ÐÑ/ Y ÐÑ/ Ð%Þ&Ñ ÐÑ ÐÑ #ÒY ÐÑ \ ÐÑÐM ÐÑÑÓÐM / 1 Ñ Then, the mean queue length at an arbitrary time can be obtained, from (24), as Pœ ÐÑ ] ÐÑ/ œ PH ÐÑ 1H \ Ð / 1 G H Ñ H/ Þ Ð%Þ'Ñ 42 The Mean Waiting Time We rewrite (10,Ñ as 5 [ÐќР) Ñ9 H < 5ÒWÐÑÓLÐÑ ) ) Ò X R5 ÐÑ ) M Ó / 5œ! Ð%Þ(Ñ R# R5# ÐZ Ñ 5 Ð 9 Ñ, ÒW Ð) ÑÓ H 5ß6 Ð) ÑL Ð) ÑÒX R56 Ð) Ñ M Ó/ 5œ! 6œ! where µ [ Ð Ñ / Rßsetup ) H Rß 5 ÐZ Ñ Z Ð) Ñ 5œ! ) [ µ setup ÐÑœ ) œ9 < 5ÒWÐÑÓLÐÑ ) ) 9, LÐÑ ) Ð%Þ)Ñ ) ÐWÑ L Ð Ñ 9, IÐLÑ ) Ð Ñ) Ò) M G W Ð) ÑHÓ Þ Then the mean waiting tie can be obtained from

22 12 HO WOO L and BOO YONG AHN R# R5# ÐÑ ÐZ Ñ [ œ [ Ð!Ñ œ Ð Ñ 9 H < 5 R5 9, H5ß6R56 5œ! 5œ! 6œ! Ð%Þ*Ñ µ ÐÑ H [ Ð!Ñ / Rßsetup where H5ß6 can be obtained from (II6) of Appendix II and can be obtained from (250) We rewrite (4) as µ Rß [ setup Ð) ÑÒ) M G W Ð) ÑHÓ œ Ð Ñœ9 < ÒW Ð) ÑÓ ) L Ð) Ñ Ð%Þ!Ñ Then, we easily get ÐZ Ñ ÐWÑ IÐLÑ 9, ÒZ Ð) ÑÓL Ð) Ñ 9, ÒL Ð) ÑÓ Þ Differentiating (410) with respect to ), we get [ µ R Ð!Ñ œ 1Þ Ð%ÞÑ µ ÐÑ µ ÐÑ [ ÐÑÒ ) ) M G WÐÑ ) H Ó[ ÐÑÒ ) M W ÐÑ ) HÓ Rßsetup Rßsetup œðñœ9 ÐÑ < ÒWÐÑÓ ) W ÐÑLÐÑ ) ) ) 9 ÐÑ < ÒWÐÑÓLÐÑ ) ) 9 < ÒWÐÑÓ ) ) L ÐÑ ) Ð%Þ#Ñ ÐZ Ñ ÐÑ ÐÑ 9, Z ÐÑLÐÑ ) ), ÒZ ÐÑÓL ) ÐÑ ) IÐLÑ ÐWÑ ÐÑ 9, L Ð) Ñ ÐÑ µ ÐÑ where 0 ÐBÑ œ 0ÐBÑ Using ) œ! in (412) and adding [ Ð!Ñ/1 to both sides yields R µ ÐÑ µ ÐÑ [ Ð!Ñ œ Ò[ Ð!Ñ/ Ó Rßsetup Rßsetup 1 Ð%Þ$Ñ ÐÑœ9 ÐZ Ñ ÐWÑ < Ð9 9 Ñ, 1ÒM IÐWÑH ÓÐ/ 1G HÑ Þ Postmultiplying (41) by H/ yields

23 Operational Behavior of the MAP/G/ Queue 1 µ ÐÑ µ ÐÑ [ Ð!ÑH/ œ [ Ð!Ñ/ Rßsetup Rßsetup Ð%Þ%Ñ ÐÑœ9 ÐZ Ñ ÐWÑ < Ð9 9 Ñ, 1ÒM IÐWÑH ÓÐ/ 1G HÑ H/ Þ Differentiating (412) once more, using ) œ! and postmultiplying / yields µ ÐÑ µ ÐÑ # [ Ð!Ñ/ œ IÐWÑ[ Ð!ÑH/ IÐW Ñ Rßsetup Rßsetup # Ð%Þ&Ñ # # ÐÑœ9 IÐWÑ ÐZ Ñ ÐWÑ IÐL Ñ < / IÐLÑ 9 IÐLÑ 9 Þ œ # #IÐLÑ From (414) and (415), we get µ ÐÑ # H IÐW Ñ [ setupð!ñ / œ 9 IÐWÑ < / IÐLÑ Rß #ÐÑ œ ÐZ Ñ # # # ÐWÑ IÐL Ñ #IÐLÑ 9 IÐLÑ 9 Ð%Þ'Ñ œ Ò IÐWÑ Ó ÐZ Ñ ÐWÑ H 1 M H 9 < Ð9 9 Ñ, Ð/ 1G H Ñ / Þ We finally using (416) in (49) to obtain the mean waiting time as # IÐW Ñ [ œ 9 IÐWÑ < / IÐLÑ #ÐÑ œ ÐZ Ñ # # # ÐWÑ IÐL Ñ #IÐLÑ 9 IÐLÑ 9 Ð%Þ(Ñ R#R5# Ð9 Ñ H ÐZ Ñ < Ð9 Ñ, H R 5ß6 R56 5œ! 6œ! œ Ò IÐWÑ Ó ÐZ Ñ ÐWÑ H 1 M H 9 < Ð9 9 Ñ, Ð/ 1G H Ñ / Þ

24 14 HO WOO L and BOO YONG AHN 5 Special Cases Let us show some special cases 51 M/G/1 Queue with Single Vacation ÐWÑ In this case, we have G œ, H œ, Rœ, 9 œ!, L Ð) Ñœ, Z! œz ÐÑ, ÐZ Ñ Z ÐÑ Z ÐDÑ 9 œ, 9 œ, 9 œ, Z ÐDÑ œ Z ÐÑ Z ÐÑ! ÐDÑ Using these in (262), (26) and (10,Ñ, we get the wellknown results, Z ÐDÑ Z ÐÑ Z ÐÑ ÐDÑ Z ÐÑ ] ÐDÑœ\ ÐDÑœ ÐÑÐDÑW ÐDÑ DW ÐDÑ Ð&Þ+Ñ Z Ð) Ñ Z Ð Ñ ) ÐÑ Z ÐÑ ) Z ÐÑ ) W Ð) Ñ [ Ð) Ñ œ Ð&Þ,Ñ 52 MAP/G/1 Queue with a Single Vacation ÐWÑ, Z ÐG Ñ /, Z ÐG Ñ! / 9 ÐZ Ñ, Z! ÐGÑ œ, Z ÐG Ñ /, <! œ, Z ÐG Ñ / and < œ!, Ð Ñ Using these in! In this case, we have Rœ, 9 œ!, LÐÑœ ), 9 œ,!! (262), (26), (10,Ñ and (41), we get, Z ÐGÑ, Z ÐG Ñ /, Z ÐG Ñ /! ] ÐDÑ œ, Z ÐDÑ!! Ð&Þ#+Ñ Ð ÑÐD ÑÐDÑÒDM ÐDÑÓ, Z ÐGÑ, Z ÐG Ñ /, Z ÐG Ñ /! \ ÐDÑ œ, Z ÐDÑ!! Ð&Þ#,Ñ ÐÑÐG DHÑÐDÑÒDM ÐDÑÓ, Z ÐGÑ Z Ð) Ñ, Z ÐG Ñ /, Z ÐG Ñ /, )! [ Ð) Ñ œ!! Ð&Þ#Ñ H Ð)) Ñ Ò M G W Ð) ÑH Ó / [ œ 9 IÐW Ñ #ÐÑ # # ÐZ Ñ # Ð&Þ#Ñ

25 Operational Behavior of the MAP/G/ Queue 15 ÐZ Ñ H š 1 ÒM IÐWÑH Ó9 < 9, Ð/ 1G H Ñ / Þ! 5 MAP/G/1 Queue with a Single Vacation and NPolicy ÐWÑ The results of this case can be obtained by simply using 9 œ! and L Ð) Ñœ in (262), (26), (10,Ñ and (41) References [1] Hwang, GU, The MAP/G/1 Queue and its Applications to ATM Networks, PhD Thesis, KAIST, Dept of Mathematics, Korea 199 [2] Kasahara, S, Takine, T, Takahashi, Y, and Hasegawa, T, AP/G/1 queues under N policy with and without vacations, J OR Soc Japan 9:2 (1996), 1212 [] Kasahara, S, Takine, T, Takahashi, Y and Hasegawa, Y, Analysis of an SPP/G/1 system with multiple vacations and limited service discipline, Queueing Sys 14 (199), 496 [4] Lee, HW, Moon, JM, and Park, JG, An analysis of the BMAP/G/1 queue I: Continuous time case, (1999), submitted [5] Lee, HW and Ahn, BY, Decompositions of the queue length distributions of the MAP/G/1 queue under multiple and single vacations with Npolicy, (1999), submitted [6] Lee, HW and Ahn, BY, Analysis of the MAP/G/1 queue with multiple vacations, N policy and setup, (1999), submitted X [] Lee, HW, Lee, SS, Park, JO, and Chae, KC, Analysis of M /G/1 queue with N policy and multiple vacations, J Appl Prob 1 (1994), [] Lucantoni, DM, New results on the single server queue with BMAP, Stoch Models :1 (1991), 146 [9] Lucantoni, DM, The BMAP/G/1 queue: A tutorial, In: Models and Techniques for Performance valuation of Computer and Communications Systems (ed by L Donatiello and R Nelson), SpringerVerlag (199), 05 [10] Lucantoni, DM, MeierHellstern, K, and Neuts, MF, A single server queue with server vacations and a class of nonrenewal arrival process, Adv Appl Prob 22 (1990), 6605 [11] Neuts, MF, A versatile Markovian point process, J Appl Prob 16 (199), 649 [12] Neuts, MF, Structured Stochastic Matrices of M/G/1 Type and Their Applications, Marcel Dekker, New York 199 [1] Niu, Z, Takahashi, Y, and ndo, N, Performance evaluation of SVCbased IPover ATM networks, IIC Trans Commun 1B:5 (199), 9495 [14] Niu, Z and Takahashi, Y, A finitecapacity queue with exhaustive vacation/closedown/setup times and Markovian arrival processes, Queueing Sys 1 (1999), 12 [15] Ramaswami, V, Th N/G/1 queue and its detailed analysis, Adv Appl Prob 12 (190), [16] Ramaswami, V, Stable recursion for the steady state vector for Markov chains of M/G/1 type, Stoch Models 4 (19), 11 [1] Strang, G, Linear Algebra and its Applications, 2nd ed, Academic Press, New York 190

26 16 HO WOO L and BOO YONG AHN [1] Takine, T and Takahashi, Y, On the relationship between queue length at a random time and at a departure in the stationary queue with BMAP arrivals, Stoch Models 14; (199), Appendix I: Representations in MatrixAnalytic Notations Since + ÐDÑßáß+ ÐDÑare eigenvectors of the matrix ÐG DHÑ, we have (Strang [1]) ÐG DHÑ œ BÐDÑH1ÖαÐDÑ B ÐDÑß (I1) 1 α ÐDÑ ÐG DHÑ œbðdñh š B ÐDÑß (I2) and ÐGHDÑB / œ ÒÐGDHÑBÓ α ÐDÑB œ BÐDÑH Ö/ B ÐDÑÞ (I) x 1 Let ÐDÑ be the matrix generating function of the number of customers that arrive during a service time Then, we have (Lucantoni, et al [10]) '! ÐGDHÑB ÐDÑ œ / WÐBÑÞ ÐI4) Using (I), we get the identity Then we get ÐDÑ œ BÐDÑH1ÖW ÐαÐDÑÑ B ÐDÑÞ (I5) 1 DW ÐαÐDÑÑ ÒDM ÐDÑÓ œ BÐDÑH š B ÐDÑÞ ÐI6) Similarly, let Z ÐDÑ be the matrix generating function of the number of customers that arrive during a vacation time Then, analogously with (I5), we get Z ÐDÑ œ BÐDÑH1ÖZ ÐαÐDÑÑ B ÐDÑÞ (I) For later use, let Z ÐDÑ be the matrix generating function of the number of customers that arrive during an elapsed (or remaining) vacation time Then, after using (I), we obtain Z ÐGDHÑB Z ÐBÑ! ÐDÑ œ ' / '! α œ BÐDÑH Ö/ B ÐDÑ ÐI) 1 ÐDÑB Z ÐBÑ

27 Operational Behavior of the MAP/G/ Queue 1 ' α ÐDÑBZ ÐBÑ œ BÐDÑ H 1 / Ÿ B ÐDÑ! Z Ðα ÐDÑÑ H 1 α ÐDÑ œ BÐDÑ š B ÐDÑÞ We note that ÐDÑ, Z ÐDÑ, ÐG DHÑ and Z ÐDÑ commute with each other From (I5) and (I), (I) becomes, in matrixanalytic notations, 1 Z ÐDÑ œ ÒBÐDÑMB ÐDÑ BÐDÑH ÖZ Ðα ÐDÑÑ B ÐDÑÓ BÐDÑ š B ÐDÑ H 1 ÐDÑ α (I9) œ ÒZ ÐDÑMÓÐGDHÑ and œ ÐGDHÑ ÒZ ÐDÑMÓ Likewise, for the setup time L, we have Þ LÐDÑ œ BÐDÑH1ÖL ÐαÐDÑÑ B ÐDÑ (I10) L L Ðα ÐDÑÑ 1 IÐLÑαÐDÑ ÐDÑ œ BÐDÑH š B ÐDÑ œ L M G H Þ (I11) Ò ÐDÑ ÓÐ D Ñ IÐLÑ Appendix II The Approach of Kasahara, et al [2] Let us define ÐH 5ß6 ÐCÑÑ 4 as the probability that, given the phase being in at the beginning of the vacation and an actual customer arrival during the vacation, 5 customers arrive during the elapsed vacation time, 6 customers arrive during the remaining vacation time, the remaining vacation time is less than or equal to C and the phase is 4 at the end of the vacation Let us define the joint matrix transform ÐD ß D ß ) Ñ as H # 5 6 C H ÐD ß D ß ) Ñ œ ' ) D D / H ÐCÑ # 56! 5œ! 6œ! # (II1)

28 1 HO WOO L and BOO YONG AHN B œ ' ' ÐGD HÑ> H ÐGD# HÑÐB>Ñ ) ÐB>Ñ B@ÐBÑ / / / > Bœ! >œ! B IÐZÑ Kasahara et al [2] showed that > Ð H ÐD ß D ß ) Ñ œ ' ' ) / / (II2) # B Bœ! >œ! >ÐB>Ñ H Ð@ M G D HÑ Ð@ M G D HÑ > œ! xx # œ max ÐGÑ They also showed that the following recursions hold J 5ß6 Ú Ðß Ñ œ Û Ü Ð@ M GÑJ5ß6Ð!ß Ñß Ð@ M GÑJ5ß6Ðß Ñ HJ5ß6Ð ß Ñß Ð œ!ñ Ð Ÿ Ÿ 5Ñ(II+Ñ HJ Ð5ß Ñß Ð œ 5 Ñ 5ß6 J 5ß6 Ú Ðß Ñ œ Û Ü J5ß6Ðß!ÑÐ@ M GÑß J5ß6Ðß ÑÐ@ M GÑ J5ß6Ðß ÑHß Ð œ!ñ Ð Ÿ Ÿ 6Ñ J Ðß 6ÑHß Ð œ 6 Ñ 5ß6 Ð II,Ñ where the matrix J 5ß6 Ðß Ñ satisfies H Ð@ M G D HÑ Ð@ M G D HÑ œ 5 6 D D J ÐßÑ (IIÑ with J!ß! Ð!ß!Ñ œ H Using (IIÑ in (II2), we get # 5ß6 5œ! 6œ! # 5 6 > Ð ÑB H ÐD ß D ß ) Ñ œ D D ' ' ) / / # 5œ! 6œ! # Bœ! B (II4) >ÐB>Ñ J Ð5ß 6Ñ>Þ œ5 œ6 xx ß ) C 5ß6 5œ! 56 # Then, we obtain the coefficient matrix H ÐÑœ ) / H ÐCÑof H ÐDßDß) Ñas

29 Operational Behavior of the MAP/G/ Queue 19 H B ) > Ð@ ) > ÐB>Ñ 5ß6 xx Bœ! >œ! œ5 œ6 Ð) Ñ œ ' ' / / J Ð5ß 6Ñ> (II5) ß Setting ) œ!, we get H 5ß6 œ # ß J ß œ5 œ6 Ð5ß6Ñ (II6) where B # ß ÐÑx! œ B (II) Appendix III Proof of Theorem From (26) and after using ÐGHK/ Ñ œ!, LK/ Ð Ñ œ / and D K ÐDÑ ± Dœ / œ, we have, œ D O ÐDÑ ± Dœ / œ F ÐGÑ K H Z ÐK ÐDÑÑ ± Dœ / (III1) D FÐGÑ K ÐG HKÑZ ÐKÑ LÐK ÐDÑÑ ± Dœ / D ( is given by œ D K ÐDÑ ± Dœ / œ ÐM K /1ÑÒM Ð/ Ñ1 Ó /, where is the mean number of customers that arrive during a service time and is given by (see Neuts [12]) Using and we get D Dœ œ ÐDÑ ± / œ / Ð/ 1G HÑ ÐMÑH/ ÑÞ Ð ÐDÑB B 5 5 D / G HK œ x ÐG HKÐDÑÑ H D KÐDÑ ÐG HKÐDÑÑ œ 5œ! 5 ÐG HK Ñ / œ!, Ð5 Ÿ #Ñß B D Dœ œ x ÐGHK ÐDÑÑB / / œ ÐG HKÑ H Þ

30 10 HO WOO L and BOO YONG AHN Thus, we get B D Dœ x! œ ' Z ÐK ÐDÑÑ / œ ÐG HKÑ H ZÐBÑÞ (III2) Multiplying (III2) by M œ Ð/1G HK Ñ Ð/1G HKÑ and using 1G Ð HK Ñ œ!, Ð #Ñ, we get Z ÐK ÐDÑÑ / D Dœ œð/1 G HK Ñ ' ÐGHKÑB ÒB/1H Ð/ MÑH ÓZÐBÑÞ! (III) Using 1H œ ÎÐ Ñ (Lucantoni et al [10]) and ' ÐGHKÑB! / ZÐBÑ œ Z ÐKÑ, we get Z ÐK ÐDÑÑ / œ / ÒZ ÐKÑM ÓÐ/1G HKÑ H (III4) D Dœ where we used Ð/1 G HK Ñ/ œ / and Ð/1 G HK Ñ / œ / Similarly, we get IÐLÑ LK Ð ÐDÑÑ / œ / ÒLK Ð ÑM ÓÐ/1GHKÑ H Þ (III5) D Dœ Using (III4) and (III5) in (III1) with 1/1 Ð GHKÑ œ 1and K/ œ / completes the proof

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