Chapter 2 Performance Analysis of Call-Handling Processes in Buffered Cellular Wireless Networks

Transcription

1 Capter 2 Performance Analysis of Call-Handling Processes in Buffered Cellular Wireless Networks In tis capter effective numerical computational procedures to calculate QoS (Quality of Service) metrics of call-andling processes in mono-service Cellular Wireless Networks (CWN) wit queues of eiter original (o-calls) or andover (-calls) calls are proposed Generalization of te results found ere for integrated voice/data CWN is straigtforward Unlike classical models of mono-service CWN, ere original and andover calls are assumed not to be identical in terms of time of radio cannel occupancy First we consider models of CWN wit queues of -calls in wic for teir prioritization a guard cannels sceme is also used We will ten consider models of CWN wit queues of o-calls and guard cannels for -calls For bot kinds of model te cases of limited and unlimited queues of patient and impatient calls are investigated For te models wit unlimited queues of eterogeneous calls te easily ceckable ergodicity conditions are proposed Te ig accuracy of te developed approximate formulae to calculate QoS metrics is sown 2 Models wit Queues for -Calls As mentioned in Cap, te main metod for prioritization of -calls is te use of reserve cannels (sared or isolated reservation) Anoter sceme for tis purpose is te efficient arrangement of teir queue in te base station However, joint use of tese scemes improves te QoS metrics of -calls Te required queue arrangement for -calls can be realized in networks were microcells are covered by a certain macrocell, ie tere exists a certain zone (andover zone -zone), witin wic mobile users can be andled in any of te neigboring cells Te time for te user to cross te -zone is called te degradation interval As te user enters te -zone a ceck is made of te availability of free cannels in a new cell If a free cannel exists, ten te cannel is immediately occupied and te -procedure is considered to be successfully completed at te given stage; oterwise te given -call continues to use te cannel of te old (previous) cell wile concurrently queuing for availability of a certain cannel of a new cell If te free cannel does not appear in te new cell before completion of te degradation interval, ten a forced call interruption of te -call occurs L Ponomarenko et al, Performance Analysis and Optimization of Multi-Traffic on Communication Networks, DOI 0007/ _2, C Springer-Verlag Berlin Heidelberg 200 3

2 32 2 Performance Analysis of Call-Handling Processes in Buffered CWN Herein we consider models of four types: (i) limited queuing of -call and infinite degradation interval; (ii) limited queuing of -call and finite degradation interval; (iii) unlimited queuing of -call and infinite degradation interval; (iv) unlimited queuing of -call and finite degradation interval In all mentioned types of model it is assumed tat a cell contains N > radio cannels and o-calls (-calls) enter te given cell by te Poisson law wit intensity λ o (λ ), te time of cannel occupancy by o-calls (-calls) being an exponentially distributed random quantity wit a mean of μ o (μ ) Note, tat te time of cannel occupancy considers bot components of occupancy time: te call service time and mobility If during te service time of any type of call te -procedure occurs, ten due to te lack of memory of exponential distribution te remaining time of te given call service in a new cell (now te -call) also as an exponential distribution wit te same mean Te different types of call are andled by te sceme of guard cannels (sared reservation), ie an entering o-call is received only wen tere exists not less tan g+ free cannels; oterwise te o-call is lost (blocked) A andover call is received wit at least one free cannel available; sould all N cannels be busy, te -call joins te queue (limited or unlimited) In all model types it is assumed tat at te moment a cannel becomes free in a new cell te queue of -calls (if it exists) is served by te FIFO (First-In-First-Out) process; wit no queue te free cannel stands idle 2 Models wit Finite Queues First we consider te models of type (i), ie models wit a limited queue of patient -calls In tis model if all N cannels are busy, ten te entering -call joins te queue wit maximal size B>, if at least one vacant place is available; oterwise (ie wen all places in te buffer are occupied) te -call is lost Since te degradation interval is infinite, te andover call cannot be lost sould it be placed in a queue In oter words, te waiting -calls are assumed to be patient Note tat tis model is adequate for networks wit slow velocity mobile users For a more detailed description of a cell s operation use is made of 2-D MC (Markov cain), ie te cell state at te arbitrary time instant is given by te vector k = (k, k 2 ), were k i is te number of o-calls (-calls) in te system, i =,2 Since o-calls are andled in a blocking mode and te system is conservative (ie wit te queue available cannel outages are not admitted), in te state k te number of -calls in cannels (k2 s) and in te queue (kq 2 ) are determined as follows: k s 2 = min {N k, k 2 }, k q 2 = (k + k 2 N) +, (2) were x + = max (0, x) Terefore, te set of all possible states of te system is determined in te following way:

3 2 Models wit Queues for -Calls 33 S := { k:k = 0,,, N g; k 2 = 0,,, N + B, k + k s 2 N, kq 2 B} (22) Note 2 In te known works in view of te calls being identical in terms of cannel occupation time, te state of a cell is described by a scalar magnitude wic points out a general number of busy cannels in a base station, ie as te matematical model one applies -D MC Since in te models studied te cannel occupation time by different types of call is different, te description of a cell by a scalar magnitude is impossible in principle Te elements of te generating matrix corresponding to 2-D MC, q(k, k ), k, k S, are determined as follows (see Fig 2): λ o if k + k 2 N g, k = k + e, λ if k = k + e 2, q(k, k ) = k μ o if k = k e, k 2 sμ if k = k e 2, 0 in oter cases (23) 08 λ 6μ 07 7 λ 6μ λ 5μ λ 5μ λ 4μ λ 4μ λ 3μ λ 3μ λ 2μ Fig 2 State transition diagram for te model wit a limited queue of patient -calls, N = 6, g = 3, B = 2 λ μ μ o 2μ o λ o λ o

4 34 2 Performance Analysis of Call-Handling Processes in Buffered CWN Hence, te matematical model of te given system is presented by 2-D MC wit a state space (22) for wic te elements of te generating matrix are determined by means of relations (23) Te stationary probability of state k is denoted by p(k) Ten in view of te model being Markovian, according to te PASTA teorem we find tat te dropping probability of -calls (P ) and probability of blocking of o-calls (P o ) are determined in te following way: P := k S p(k)δ ( k q 2, B), (24) P o := k S p(k)i(k + k2 s N g) (25) Te average number of busy cannels of te cell (Ñ) and te average lengt of te queue of -calls (L ) are also determined via stationary distribution of te model: were Ñ := L := N jς (j), (26) j= B lτ (l), (27) l= ς (j) := p (k) δ ( k + k2 s, j) and τ (l) := p (k) δ ( k q 2, l) k S k S are te marginal distribution of a model Hence, to find QoS metrics (24), (25), (26), and (27) it is necessary to determine te stationary distribution of te model p(k), k S, from te corresponding system of global balance equations (SGBE) Tis is of te following form: p (k) ( λ o I ( k + k2 s g ) ( ( q + λ δ k 2, B) + ( k + k2 s ) )) μ = λ o p (k e )( δ (k,0)) I ( k + k s 2 g 2) + λ p (k e 2 )( δ (k 2,0)) + (k + ) μp (k + e ) I (k < N g ) + (k2 s + )μp (k + e 2) I ( k2 s < N), k S, (28) p (k) = (29) k S However, to solve te last problem one requires laborious computation efforts for large values of N and B since te corresponding SGBE (28) and (29) ave no explicit solution As was mentioned in Sect 2, very often te solution of suc

5 2 Models wit Queues for -Calls 35 problems is evident if te corresponding MC as a reversibility property and ence tere exists a stationary distribution for it of te multiplicative type However, by applying Kolmogorov criterion for 2-D MC one can easily demonstrate tat te given MC is not reversible Indeed, te necessary reversibility condition states tat if te transition from state (i, j) into te state (i, j ) exists, ten te reverse transition also exists However, for te MC considered tis condition is not fulfilled So by te relations (23) in te given MC te transition (k, k 2 ) (k, k 2 ) wit intensity k μ o exists, were k +k 2 >N g, wile te inverse transition does not exist To overcome tese difficulties one suggests employing te approximate metod of calculating te stationary distribution of te 2-D MC It is acceptable for models of micro- and picocells for wic te intensity of -calls entering greatly exceeds tat of o-calls and te talk time generated by an -call is sort In oter words, ere it is assumed tat λ >> λ o, μ >> μ o (for respective comments see Sect 2) Consider te following splitting of te state space (22): were S = N g j=0 S j, S j Sj =, j = j, (20) S j := {k S : k = j}, j = 0, N g Te sets S j are combined in merged state < j > and te merging function wit te domain (22) is introduced: U(k) =< j > if k S j, j = 0, N g (2) Te merging function (2) specifies te merged model, wic is -D MC wit state space S := { < j >: j = 0, N g } Te above assumption about relations of loading parameters of different call types provides te fulfillment of conditions necessary for correct application of pase-merging algoritms: intensities of transitions between states inside eac class S j,j= 0,,,N g,essentially exceed intensities of transitions between states from different classes To find te stationary distribution of te initial model one needs a preliminary determination of te stationary distribution of split and merged models Te stationary distribution of te jt split model wit space of states S j is denoted by ρ j (i), j = 0,,, N g, i = 0,,, N +B j, ie ρ j (i) is te stationary probability of state (j, i) S j in jt split model It is determined as te stationary distribution of classical queuing system M/M/N j/b wit te load ν Erl, ie ν i ρ j (i) = i! ρj (0) if i =, N j, ν i (N j)!(n j) i+j N ρj (0) if i = N j +, N j + B, (22)

6 36 2 Performance Analysis of Call-Handling Processes in Buffered CWN were N j ρ j (0) = v i i! + (N j)! N j+b i=n j+ v i (N j) i+j N To find te stationary distribution of te merged model one sould preliminarily determine te elements of te generating matrix corresponding to -D MC denoted by q(< j, < j >), < j >, < j > S Te following relations determine te mentioned parameters: were λ o (j + ) if j = j +, q(< j >, < j >) = j μ if j = j, 0 in oter cases, N g i (i + ) = ρ i (0) j=0 v j, i = 0, N g j! (23) Te last relations imply te stationary distribution of te merged model π(< j >), < j > S, to be determined as te corresponding distribution of te appropriate birt-and-deat process In oter words π(< j >) = vj o j! j (i)π(< 0 >), j =, N g, (24) i= were π(< 0 >) = + N g i= v i o i! i (j) j= Summing up everyting stated above one can offer te following approximate formulae for calculating te QoS metrics (24), (25), (26), and (27) of te given model: N g P ρ j (N + B j)π(< j >); (25) P o j=0 N g N+B j j=0 i=n g j ρ j (i)π(< j >); (26)

7 2 Models wit Queues for -Calls 37 Ñ N g j= j j ρ i (j i)π(< i >) + N j j=n g+ N g ρ i (k i)π(< i >) + N N g N+B i j=n i ρ i (j)π(< i >); (27) L N+B g j= j g ρ i (j)π(< i >) + N+B j=n+b g+ N+B j j ρ i (j)π(< i >) (28) Now we can generalize te obtained results to te model wit a limited queue of -calls wit a finite degradation interval [ie to te model of type (ii)] Unlike te previous model in tis model it is assumed tat te degradation interval for -calls are independent, equally distributed random quantities aving exponential distribution wit te finite mean γ Te state space of te given model is defined by (22) also But for te given model te elements of te generating matrix corresponding to 2-D MC, q(k, k ), k, k S are determined as follows (see Fig 22): λ o if k + k 2 < N g, k = k + e, λ if k = k + e 2, q(k, k ) = k μ if k = k e, k 2 sμδ (k 2 q,0) + (k s 2 μ + k q 2 γ )( δ (k q 2,0)) if k = k e 2, 0 in oter cases (29) Te stationary probability of blocking of o-calls (P o ) in tis model is determined also by te formula (25) However, in tis model losses of -calls occur in te following cases: (a) If at te moment of its entering te queue tere are already B calls of te given type; (b) If te interval of its degradation is completed before it gains access to a free cannel Hence te given QoS metric is determined as follows: P := k S p(k)δ ( k q 2, B) + λ B N g iγ p (j, N + i j) (220) i= j=0 In te last formulae te first term of te sum defines te probability of te event corresponding to case (a) above, wile te second term of te sum defines te probability of even corresponding to case (b) above For te given model te average number of busy cannels of te cell (Ñ) and te average lengt of te queue of -calls (L ) are also determined analogously to

8 38 2 Performance Analysis of Call-Handling Processes in Buffered CWN 08 λ 6μ + 2γ λ 07 λ 6μ + γ 7 5μ + 2γ λ 5μ + γ λ 4μ + 2γ λ 4μ + γ λ 3μ + 2γ λ 3μ + γ λ 2μ λ μ μ o 2μ o λ o λ o Fig 22 State transition diagram for te model wit a limited queue of impatient -calls, N = 6, g = 3, B=2 (26) and (27) And SGBE for te given model is also derived in a similar way to (28) and (29) However, as in te model wit patient -calls, to solve te indicated SGBE requires laborious computation efforts for large values of N and B since te corresponding SGBE as no explicit solution But to overcome tese difficulties use is made of te approximate metod proposed above for te model wit patient -calls In tis case te splitting (20) of state space (22) is considered also { and by function (2) } an appropriate merged model wit state space S := < j >: j = 0, N g is constructed But in tis case te stationary distribution in te jt split model is determined as follows: v i ρ j i! ρj (0) if i =, N j, (i) = i+j N v N j (N j)! l= λ l (N j) μ + lγ ρj (0) if i = N j +, N j + B, (22)

9 2 Models wit Queues for -Calls 39 were N j ρ j v (0) = i i! + vn j (N j)! N j+b i=n j+ i+j N l= λ l (N j) μ + lγ, j = 0, N g Furtermore, te stationary distribution of te merged model π(< j >), < j > S is determined in a similar way to (24) Note tat in tis case te terms in (23) and (24) are calculated by taking into account formulae (22) Making use of te above results and omitting te known intermediate matematical calculations te following formulae to calculate te QoS metrics of a network wit a limited queue and finite interval of degradation of -calls are obtained: N g P ρ i (N + B i)π(i) + λ B N g iγ ρ j (N + i j) π (j); (222) i= j=0 P o N g N+B j j=0 i=n g j ρ j (i)π(< j >); (223) Ñ N g j= j j ρ i (j i)π(< i >) + N j j=n g+ N g ρ i N g (j i)π(< i >) + N j=0 N+B j π(< j >) i=n j ρ j (i); (224) L N+B g j= j g ρ i (j)π(< i >) + N+B j=n+b g+ N+B j j ρ i (j)π(< i >) (225) 22 Models wit Infinite Queues We now consider te model of type (iii), ie a model of a cell wit an unlimited queue of -calls and an infinite degradation interval Te set of all possible states of te given model is determined in te following way: S := { k:k = 0,,, N g; k 2 = 0,, ; k + k s 2 N} (226) Te number of -calls in te queue and in cannels and te elements of te generating matrix are calculated in a similar way to (2) and (23), respectively (see Fig 23) Te required QoS metrics in tis case is also calculated via a stationary distribution of te model tat is determined from te corresponding SGBE of infinite

10 40 2 Performance Analysis of Call-Handling Processes in Buffered CWN μ 7 6μ 5μ μ 4μ μ 3μ μ μ μ μ 2μ o o o o Fig 23 State transition diagram for te model wit an unlimited queue of patient -calls, N = 6, g = 3 dimension However, te employment of te metod of two-dimensional generating functions for finding te stationary distribution of te given model from te mentioned SGBE is related to well-known computational and metodological difficulties In relation to tis we sall apply te above-described approac to calculating te stationary distribution of te model Witout repeating te above procedures we just note tat ere we also made use of te split sceme of te state space (226) analogous to (20) Since te selection of te split sceme completely specifies te structures of te split and merged models, below only minor comments are made on te formulae suggested

11 2 Models wit Queues for -Calls 4 Te stationary distribution of te jt split model is determined as te stationary distribution of te classical queuing system M/M/N j/ wit te load ν Erl, ie were ρ j (0) = v i ρ j i! ρj (0), i =, N j, (i) = v i (N j) N j i ρ j (0), i N j, (N j)! N j v i i! + v N j (N j)! N j N j v, j = 0,,, N g (227) Te ergodicity condition of te jt split model is ν <N jhence, for te stationary mode to exist in eac split model te condition ν <gsould be fulfilled Note tat te model ergodicity condition is independent of te o-calls load Tis sould ave been expected since by te assumption λ >> λ o, μ >> μ o o-calls are andled by te sceme wit pure losses In te particular case g = we obtain te condition ν < Te stationary distribution of te merged model is π(< j >), < j > S determined in a similar way to (24) However, in tis case one sould consider te fact tat in te above formulae te parameters ρ j (0), j = 0,,, N g, are calculated from (227) After performing te necessary matematical transformations one obtains te following approximate formulae for calculating te QoS metrics of te model wit an unlimited queue of patient -calls and guard cannels available: Ñ N g j= j P o N g j=0 j ρ i (j i)π(< i >) + ( N g + N π (< j >) j=0 N g L N j N g j N j=n g+ ) ρ j (i) ; ρ j (i) π (< j >); (228) ρ i (k i)π(< i >) N g j π (< i >) ρ i (0) vn+ i (N i)! (229) N i (N i v ) 2 (230) Now we consider te model of type (iv), ie a model wit an unlimited queue of impatient -calls wit te guard cannels available In te given model te impatient -call can be lost from te unlimited queue unless a single cannel of a new cell is

12 42 2 Performance Analysis of Call-Handling Processes in Buffered CWN μ μ + 5μ μ + 4μ μ + 3μ μ μ μ μ o 2μ o o o Fig 24 State transition diagram for te model wit an unlimited queue of impatient -calls, N = 6, g = 3 freed before termination of its degradation interval To obtain tractable results as in te model of type (ii) it is assumed tat degradation intervals for all -calls are independent, equally exponentially distributed random quantities wit te finite mean γ Te state space of tis model is given by means of te set (226) However, tereby te elements of te generating matrix of te corresponding 2-D MC are determined in a similar way to (29) (see Fig 24) Similarly to te previous model for te given model one can develop te appropriate SGBE for stationary probabilities of te system states However, te above-mentioned difficulties of applying suc SGBE in tis case are even more complicated since to solve tis one te approximate approac is used

13 2 Models wit Queues for -Calls 43 Te stationary distribution of te jt split model in te given case is determined by v i ρ j (i) = v N j (N j)! were N j ρ j (0) = i! ρj (0), ifi =, N j, i v i i! k=n j+ + v N j (N j)! λ (N j) μ + (k + j N) γ ρj (0), ifi N j +, m m=n j+ k=n j+ λ (N j) μ + (k + j N) γ (23) It is wort noting tat in te given model at any permissible values of load parameters in te system tere exists a stationary mode It can be easily proved since te analysis of te ratio limit of two neigboring terms of a series sows tat te numerical series R := m m=n j+ k=n j+ λ (N j) μ + (k + j N) γ (232) involved in determining ρ j (0) [see formulae (23)] converges at any positive values of load parameters of -calls and degradation interval However, unfortunately one does not manage to find te exact value of te sum of series (232) but we manage to find te following limits of tis sum cange: ( exp λ (N j) μ + γ ) R exp ( λ γ ) (233) Note 22 From te last relations one can see tat in a particular case, wen (N j) μ << γ or te quantity (N j) μ is sufficiently small, te approximate value of sum R can be used as te rigt-and side of inequality (233) Te QoS metrics of te given model are determined like (228), (229), and (230) but now te described formulae involve te corresponding distributions for te model wit impatient -calls In te given model losses of -calls from te queue due to teir impatience also occur Te probability of suc an event is calculated as follows: P = λ n= N g nγ p (i, N + n i) (234)

14 44 2 Performance Analysis of Call-Handling Processes in Buffered CWN 23 Numerical Results Te algoritms suggested allow one to study te beavior of QoS metrics of te systems investigated in all admissible ranges of canges to teir structural and load parameters In order to be sort, among te models of cells wit a queue of -calls, only te results for models wit unlimited queues are given in detail Some results of numerical experiments for te model wit patient -calls at N = 5 and μ o = 05 are sown in Figs 25, 26, and 27 Tey completely confirmed all teoretical expectations So, te probability of losing o-calls grows (see Fig 25) and te average number of busy cannels (see Fig 26) and te average number of -calls in te queue (see Fig 27) falls as te number of guard cannels increases As canges to te average time of -call delay coincide wit te analogous dependence on teir average number in a queue, te details of tis function are not presented ere Note tat all te functions under study are increasing wit respect to intensity of o-call traffic However, unlike te function Ñ te rates of cange of te functions P o and L are sufficiently ig So, at N = 5, ν o = 4, and ν = 08 te values of functions P o and L at te points g = and g = 0 equal P o () = 38E-04, P o (0) = 28E-0, L () = 99E-05, L (0) = 2E-0, and tose of te function Ñ at tese points equal and 3655 Te analysis of results of numerical experiments sows tat regardless of te essential difference in loads of o-calls, as te number of guard cannels grows te corresponding values of functions P o and L become closer For example, in two experiments at N = 5 te load parameters were selected in tis way: () ν o = 4, ν = 08; (2) ν o = 2, ν = 075 For tis data te following ratios old: P o ()/P o 2 () 400, P o (4)/P o 2 (4) 0; L ()/L 2 () 0 3,L (4)/L 2 (4) 3, LgP o 2 Fig 25 P o versus g for te model wit an unlimited queue of -calls in te case were N = 5; μ o = 05; λ o = 2, λ = 4 μ = 5; 2 λ o =, λ = 3 μ = 4 g

15 2 Models wit Queues for -Calls 45 2 Fig 26 Ñ versus g for te model wit an unlimited queue of -calls in te case were N = 5; μ o = 05; λ o = 2, λ = 4 μ = 5; 2 λ o =, λ = 3 μ = 4 g LgL 2 Fig 27 L versus g for te model wit an unlimited queue of -calls in te case were N = 5; μ o = 05; λ o = 2, λ = 4 μ = 5; 2 λ o =, λ = 3 μ = 4 g were P o i (L i ) is te value of te function P o (L ) in te it experiment, i =,2 Te corresponding ratios for te function Ñare of te form Ñ ()/Ñ 2 () 8,Ñ (4)/Ñ 2 (4) Te results of numerical experiments for te model wit impatient -calls sowed tat te probability of losing o-calls also grew as te number of guard cannels increased In tis model te probability of losing -calls decreases wit respect

16 46 2 Performance Analysis of Call-Handling Processes in Buffered CWN to te number of guard cannels and te increase in te number of guard cannels also decreases te average number of busy cannels Bot functions increase wit respect to te intensity of o-call traffic Te analysis of QoS metrics of different models wit equal initial data sowed tat in te model wit patient -calls te probability of losing o-calls was iger tan in te model wit impatient -calls Tis sould ave been expected since in te model wit impatient -calls tere occur losses of -calls from te queue tereby increasing te cances for o-calls to occupy a free cannel Tese comments also refer to oter QoS metrics namely utilization of cannels in te model wit impatient -calls is worse tan in te model wit patient -calls And te average lengt of te queue of -calls in te model wit patient -calls is larger tan tat in te model wit impatient -calls Anoter goal of performing numerical experiments was te estimation of te proposed formula accuracy Te exact values (EV) of te QoS metrics for te model wit patient -calls at te identical time of cannel occupancy by different types of calls are determined by te following formulae wic are easily derived from classical one-dimensional birt-and-deat processes: were N g P o = ρ k, k=0 ( ) N N Ñ = kρ k + N ρ k, k= L = AṽN+ ( ṽ ) 2, k=0 v k ρ k = k! ρ 0 if k =, N g, ( ) N g λλ v k k! ρ 0 if k = N g +, N, N g ρ 0 = v k ( ) λ N g k! + N v k ( ) λ N g λ k! + NN λ N! k=0 k=n g+ ṽ N+ ṽ λ := λ o + λ ; v := λ μ ; μ := μ o = μ ; ṽ := v ( ) λ N g N ; A := NN λ N! Note tat approximate values (AV) of QoS metrics are almost identical to teir exact values wen te accepted assumption about ratios of load parameters of o- and -calls is valid Some comparisons for te models wit parameters N = 0, ;

17 2 Models wit Queues for -Calls 47 λ 0 = 03, λ = 2, μ o = μ = 3 and N = 5, λ 0 = 2, λ = 4, μ o = μ = 5 are presented in Tables 2 and 22, respectively As indicated in Tables 2 and 22 te accuracy of approximate formulae is sufficiently ig even wen te accepted assumption is not valid, ie μ o = μ Similar results are obtained for any possible values of initial data of te model It is wort noting tat sufficiently ig accuracy exists for te initial data not satisfying te above-mentioned assumption concerning te ratios of traffic of o- and -calls In oter words, te numerical experiments for te models wit symmetric traffic (ie λ o = λ ) in addition to te ones in wic te intensity of o-calls greatly exceeds te intensity of -calls sowed a sufficiently ig accuracy for te suggested approximate formulae Certain results of te comparison for te models Table 2 Comparison of exact and approximate values of QoS metrics for te model wit patient -calls in te case were N = 0, λ o = 03, λ = 2, μ o = μ = 3 P o Ñ L g EV AV EV AV EV AV 2556E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-0 Table 22 Comparison of exact and approximate values of QoS metrics for te model wit patient -calls in te case were N = 5, λ o = 2, λ = 4, μ o = μ = 5 P o Ñ L g EV AV EV AV EV AV E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-5

18 48 2 Performance Analysis of Call-Handling Processes in Buffered CWN wit parameters N = 0, λ o = 2, λ = 03, μ o = μ = 3 and N = 5, λ o =λ = 4, μ o = μ = 5 are sown in Tables 23 and 24, respectively As seen from Tables 23 and 24 small deviations are observed upon calculation of average lengt of -calls Terewit te approximate values of te QoS metrics are always larger tan teir exact values Te last circumstance suggests tat to increase te reliability of network performance te obtained approximate formulae can be applied at te initial stages of its design Analogous results are obtained for te models wit a limited queue of -calls For te model wit a limited queue of patient -calls te approximate results obtained were compared wit results in [], werein accurate formulae for te model wit identical (wit respect to cannel occupation times) calls were developed Te Table 23 Comparison of exact and approximate values of QoS metrics for te model wit patient -calls in te case were N = 0, λ o = 2, λ = 03, μ o = μ = 3 P o Ñ L g EV AV EV AV EV AV 8332E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-7 Table 24 Comparison of exact and approximate values of QoS metrics for te model wit patient -calls in te case were N = 5, λ o = λ = 4, μ o = μ = 5 P o Ñ L g EV AV EV AV EV AV 76280E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-5

19 22 Models wit Queues for o-calls 49 comparative analysis was performed over a wide range of variance in te initial data of te model It is wort noting tat te approximate and exact results from calculating te loss probability of -calls were identical An insignificant difference is observed in calculating te blocking probability of o-calls It can be seen tat te maximal value of error in all executed experiments was negligible and in most cases equals zero even for data not satisfying te above-mentioned assumption concerning te ratios of traffic of o- and -calls So, for example, at N = 50,B= 0, λ 0 = 2, λ =, μ = 0 te maximal error olds for g = 49 and is 6%, ie for tese initial data te exact value of P o equals 0233 wile its approximate value equals Models wit Queues for o-calls In order to compensate for te cance of o-calls a queue (limited or unlimited) is required for tem, wile maintaining a ig cance for -calls to access te system via reservation of cannels Here we consider four types of models wit queues for o-calls wereas -calls are treated according to a lost model [2, 9] Note tat in tis section it is assumed tat te required andling time is independent of call type and exponentially distributed wit te same average μ In all scemes it is assumed tat all m+n cannels are divided into two groups: a Primary group wit m cannels and a Secondary group wit n cannels If all cannels in bot groups are busy te -call is lost New calls are only allowed to te Primary group; terefore if all m cannels are busy tis call is placed into a queue In two scemes reallocation of cannels from one group to anoter is not allowed, ie isolated reservation is considered In te NOPS (Handoff calls Overflow from Primary to Secondary) sceme, for andling of an -call, te Primary group cannels are used first and upon absence of an empty cannel in te Primary group (all m cannels are busy) te Secondary group cannels are used In te HOSP (Handoff calls Overflow from Secondary to Primary) sceme te searc for a free cannel for andling of an -call is realized in te Secondary group first Te last two scemes involve sared reservation of cannels Tey maybe described as follows In one sceme upon release of a cannel in te Primary group (eiter by an o-call or -call) an -call from te Secondary group is reallocated to te Primary group regardless of te lengt of te o-calls in te queue However, o-calls from te queue are admitted to cannels in accordance wit te FIFO discipline only if te number of total empty cannels exceeds n Te given cannel reallocation sceme is called Handoff Reserve Margin Algoritm (HRMA) Te main difference of te final sceme from te previous one is in te reallocation of te -call from te Secondary group to te Primary group Upon release of a cannel in te Primary group (eiter by an o-call or -call) an -call from te Secondary group is reallocated to te Primary group if and only if tere are no o-calls in te queue In oter words reallocation of -calls from te Secondary group to te Primary group is not allowed until te queue does not contain any o-calls Tis reservation sceme is called Handoff calls Overflow from Primary to Secondary wit Rearrangement (HOPSWR)

20 50 2 Performance Analysis of Call-Handling Processes in Buffered CWN 22 Models Witout Reassignment of Cannels Here we consider two isolated reservation scemes in wic reallocation (reassignment) of cannels from one group to anoter is not allowed In te NOPS (Handoff calls Overflow from Primary to Secondary) sceme te initial searc for a free cannel for service of an -call is carried out in te Primary Group and upon te absence of an empty cannel in te Primary group (all m cannels are busy) Secondary group cannels are used (see Fig 28a) In te HOSP (Handoff calls Overflow from Secondary to Primary) sceme te initial searc for a free cannel to service an -call is carried out in te Secondary Group (see Fig 28b) In bot scemes o-calls can be served only in te Primary Group of cannels and any conservative discipline of service wic does not admit to idle times of cannels in te presence of a queue can be used for service of te o-calls queue In bot scemes in cases of occupancy of all m+n cannels te -call is lost and any reassignment of te -call from one group to anoter is not permitted First of all we sall consider te HOPS sceme in te model wit an infinite queue of o-calls Te QoS metrics include probability of loss of -calls (P ), average lengt of o-calls queue (L o ), and te average latency period in te queue (W o ) Te following ierarcical approac can be used for te analysis of tis model as in tis sceme received -calls initially go to te Primary Group of cannels, and o queue I II m n exit (a) Fig 28 Diagram of te HOPS (a) and HOSP (b) model o queue II (b) I m n exit

21 22 Models wit Queues for o-calls 5 Fig 29 Hierarcical approac for study of te HOPS model (a) o queue exit m P (b) P exit n P only missed calls of te given type are furter received in te Secondary Group of cannels In te first step of te ierarcy (see Fig 29a) we sall consider a system wit m cannels wic serve calls of two types wit rates λ o and λ, tus te olding time of any type of call as an exponential distribution wit a common mean μ New calls are buffered in an infinite queue and -calls are lost in case of occupancy of all cannels Missed -calls are forwarded to te Secondary Group of cannels for service Consider tat te probability of loss of -calls is equal to P in te Primary Group of cannels From Poisson flow property we deduce tat te input to te Secondary Group of cannels forms Poisson flow wit rate λ := λ P Hence on te second stage of ierarcy (see Fig 29b) we examine classic Erlang model M/M/n/0 Loss probability of calls of tis model ten will be considered as te desired P Andte desired L o and W o are calculated as QoS parameters of queuing system described at te first step of ierarcy Now we sall consider te problem of calculation of te above-specified QoS metrics Te state of te queuing system wit two types of calls described in te first step of te ierarcy is given by scalar parameter k wic specifies te total number of calls in te system, k = 0,,2, Stationary distribution of te corresponding one-dimensional birt-deat process (-D BDP) is calculated as (Fig 20): v k ρ k = k! ρ 0 if k m, v m m! vk m o m k m ρ 0 if k m +, (235)

22 52 2 Performance Analysis of Call-Handling Processes in Buffered CWN o o o 0 m m+n μ 2μ mμ (m+)μ (m+n)μ (m+n)μ Fig 20 State transition diagram for te HOPS model were v := λ/μ, λ := λ o + λ, v o := λ o /μ, ρ 0 = ( m k=0 v k k! + vm m! v o m v o ) On derivation of formulae (235) an intuitively clear and simple condition of ergodicity of te models becomes apparent: v o < m Hence we can see tat te condition of ergodicity of te models does not depend on te loading of andover calls From (235) it is concluded tat te probability of -call loss in te Primary Group (P ) is defined as: m P = ρ k (236) Hence, required QoS metric P is calculated by means of Erlang s classical B- formula for te M/M/n/0 c model wit a load of ṽ := λ /μ Erl In oter words, P = E B (ṽ, n) After certain transformations we obtain te following formula for calculation of QoS metric L o : L o = kρ k+m = k= k=0 v m (m )! v o (m v o ) 2 ρ 0 (237) QoS metric W o is obtained from te Little formulae, ie W o = L o /λ Te developed approac allows te definition of QoS metrics of te model also in te presence of a limited buffer for waiting in a queue of o-calls We sould note tat in tese models at any loading and structural parameter values in te system tere is a stationary mode, ie ergodicity performance is not required, v o < m Let te maximum size of te buffer be equal to R, R< On te basis of formulae for finite BDP we conclude tat stationary distribution of te appropriate system is calculated as follows: ρ k ν k ρ 0 = k! ν m νo m! m k m k m если k m, (238) ρ 0 если m + k m + R,

23 22 Models wit Queues for o-calls 53 were m ρ 0 = v k k! + vm m! mm v m o k=0 R k=m+ ( vo m ) k Ten by using (238) from (236) and Erlang s B-formula QoS metric P is calculated And from (238) te following formula for calculation of te QoS metric L o (R) for te model wit a limited queue of o-calls finds: L o (R) = R kρ k+m (239) From (239) quantity W o (R) in te given model is calculated as follows W o (R) = k= L o (R) λ o ( P o (R)), were P o (R) denotes te loss probability of o-calls in te given model, ie P o (R) = ρ m+r Now consider te HOSP sceme in te model wit an infinite queue of o-calls As in te previous sceme it is possible to use te ierarcical approac Here in te first step of te ierarcy (see Fig 2a) te classical Erlang model M/M/n/0 wit (a) exit n (b) o queue exit Fig 2 Hierarcical approac for study of te HOSP model m

24 54 2 Performance Analysis of Call-Handling Processes in Buffered CWN a load v := λ /μ Erl is considered Te probability of -call loss in tis model will be denoted by P 2, ie P 2 = E B (v, n) Missed -calls in tis system are forwarded to te Primary Group for reception of service Hence, in te second step of te ierarcy (see Fig 2b) te system wit m cannels wic serves calls of two types wit rates λ o and ˆλ := λ P 2, tus te olding time of a call of any type tat as an exponential distribution wit te general average μ, is considered New calls are buffered in an infinite queue and in te case of occupancy of all cannels -calls are lost Missed -calls in tis system are finally lost Tus, stationary distributions of te queuing system described in te second step of te ierarcy are calculated by means of te formula (235) However, in tis case te specified formula parameter λ is determined as λ = λ o + ˆλ Stationary distribution of te last system will be denoted troug σ k, k = 0,,2, Te condition of ergodicity of models in te given sceme is also v o < m Ten in view of te above-stated, it is concluded tat te required QoS metric P in te HOSP sceme of cannel distribution is calculated as: m P = σ k (240) Oter QoS metrics L o and W o in te given sceme of cannel allocation are also calculated from (237) and Little s formula, accordingly Tus, it is necessary to consider tat in tis case λ = λ o + ˆλ Te above-described approac migt also be used for calculation of te QoS metrics in te HOSP sceme wit a limited queue of o-calls As tis procedure is almost te same as te analogous procedure for te HOPS sceme, it is not presented ere k=0 222 Models wit Reassignment of Cannels In tis section we consider two scemes wit reassignment of cannels First we consider te HRMA (Handoff Reserve Margin Algoritm) sceme for cannel assignment In tis case for te andling of an -call a cannel from te Primary group is used first and upon absence of an empty cannel in tis group (all m cannels are busy) Secondary group cannels are used If all cannels in bot groups are busy te -call is lost New calls are only allowed to te Primary group; terefore if all m cannels are busy tis call is placed into a queue Upon release of a cannel in te Primary group (eiter by an o-call or -call) an -call from te Secondary group is reallocated to te Primary group regardless of te lengt of te o-calls in te queue However, o-calls from te queue are admitted to cannels in accordance wit FIFO discipline only if te number of total empty cannels exceeds n Te system s state at any time is described by te two-dimensional vector k= (k, k 2 ), were k is te total number of o-calls in te system k = 0,,2,, and k 2

25 22 Models wit Queues for o-calls 55 is te total number of busy cannels, k 2 = 0,,, m+n Note tat state space S of te given model does not include vectors k wit components k >0, k 2 <m On te basis of te adopted cannel allocation sceme, we can conclude tat elements of te generating matrix q (k, k ), k, k S of te appropriate 2-D MC are defined from te following relations (see Fig 22): λ o + λ, if k = 0, k 2 m, k = k + e 2, λ, if k 2 m, k = k + e 2, q(k, k λ ) = o, if k 2 m, k = k + e, k 2 μ, if k 2 = m, k = k e 2, mμ, if k 2 = m, k = k e, 0 in oter cases (24) Te desired QoS metrics for te model wit an infinite queue of o-calls are calculated by stationary distribution of te model as follows: P = L o = p (k, m + n), (242) k =0 m+n k = k 2 =m k p (k, k 2 ), (243) Here we develop simple approximate computational procedures to calculate tese QoS metrics For correct application of te approximate approac condition λ 0 +λ λ 0,0 0 +λ λ λ λ 0,m 0,m 0,m+ 0,m+n µ (m )µ mµ (m+)µ (m+n)µ λ 0 mµ λ 0 λ 0 λ λ,m,m+,m+n (m+)µ (m+n)µ λ 0 mµ λ 0 λ 0 λ λ 2,m 2,m+ 2,m+n (m+)µ (m+n)µ λ 0 mµ λ 0 λ 0 Fig 22 State transition diagram for te HRMA model wit an unlimited queue of patient o-calls

26 56 2 Performance Analysis of Call-Handling Processes in Buffered CWN λ >> λ o is required As was mentioned in previous sections tis condition is true for micro- and picocells In addition, we assume tat bot types of calls are caracterized by very sort duration wit respect to teir frequency of arrivals Te last assumption is valid for many real system models tat are quite close to tose in tis work (see Sect ) Te following splitting of state space of te given model is considered: were S = S i, S i Sj =, i = j, (244) S i := {k S : k = i} Furtermore, te class of microstates S i is merged into te isolated merged state < i > and an appropriate merged model wit state space S := {< i >: i = 0,, 2, } is constructed Te elements of te generated matrix of splitting models wit state space S i tat is denoted by q i (k, k ), k, k S i are calculated as follows [see (24)]: For te model wit state space S 0 : λ o + λ, ifk 2 m, k ( q 0 k, k ) 2 = k 2 +, λ =, ifm k 2 m + n, k 2 = k 2 +, k 2 μ, ifk 2 = k 2, 0 in oter cases, (245) For models wit state spaces S i, i : ( q i k, k ) λ if k 2 = k 2 +, = k 2 μ if k 2 = k 2, 0 in oter cases (246) By using (245) and (246) te stationary distribution of te splitting models are calculated from te following expressions: For i = 0: For i>0: ρ 0 (j) = v j j! ρ 0 if j =, m, ( v v ) m v j j! ρ 0 if j = m +, m + n (247) ρ i (j) = m! v m v j j! ρ, j = m +, m + n, (248)

27 22 Models wit Queues for o-calls 57 were m ρ 0 = v i ( ) v m m+n i! + v i=m+ v i i!, ρ = ( m! v m m+n i=m ) v i, v := v o + v i! By using (245), (246), (247), and (248) we conclude tat te elements of te generating matrix of merged model q ( < i >, < i > ), < i >, < i > S are calculated as follows: m+n λ o ρ q ( < i >, < i > ) 0 (j) if i = 0, i = i +, j=m = λ o if i 0, i = i +, (249) mμρ if i 0, i = i, 0 in oter cases From (249) we find te following ergodicity condition of te merged model: or in explicit form a := ( v o m m! m+n v m i=m v o mρ (m) < ) v i < (250) i! Note 23 It is important to note tat ergodicity condition (250) is exactly te stability condition of te system establised in [2], ie ere we easily obtain te stability condition of te investigated model By fulfilling te condition (250) te stationary distribution of te merged model (π(< i >) :< i > S) is calculated as were π(< i >) = a i bπ(< 0 >), i =, 2,, (25) m+n b := i=m ρ 0 (i), π(< 0 >) = a a + ab In summary te following simple formulae for calculation of te desired QoS metrics (242) and (243) can be suggested: P a + ab (( a)e B(v, m + n) + abeb T (v, m + n)), (252) ab L o ( a + b)( a), (253)

28 58 2 Performance Analysis of Call-Handling Processes in Buffered CWN were E T B (v, m + n) te truncated Erlang s B-formula, ie E B T (v, m + n) := v m+n ( m+n (m + n)! i=m ) v i i! From (252) we conclude tat P is te convex combination of two functions E B (v, m+n) and E T B (v, m+n) In oter words, for any admissible values of number of cannels and traffic loads te following limits for P may be proposed: E B (v, m + n) P E T B (v, m + n) (254) Te limits in (254) will be acieved and are te same only in te special case m = 0, ie wen only -calls arrive in te system Te proposed approac also allows calculation of QoS metrics for te model wit a limited queue of o-calls Indeed, let te maximal lengt of te queue of o-calls be R, R< Ten for any admissible values of number of cannels and traffic loads in tis system te stationary mode exists, ie in tis case fulfilling of te ergodicity condition (250) is not required For te given model te number of splitting models is R+ and teir stationary distribution are calculated by (247) and (248) Making use of te above-described approac and omitting te known transformation te following expressions are determined to calculate te stationary distribution of te merged model: were π(< i >) = a i bπ(< 0 >), i =, R, (255) π(< 0 >) = ( + ab ) ar a Since te approximate values of te QoS metrics for te model wit a limited queue of o-calls are calculated as follows: P (R) π(< 0 >)E B (v, m + n) + ( π(< 0 >))E T B (v, m), (256) L o (R) ab ar (R + + Ra) ( a) 2 π (< 0 >), (257) L o (R) W o (R) λ o ( P o (R)), (258) were P o (R) is te probability of loss of o-calls wic is calculated by P o (R) π(r)orp o (R) a R bπ(< 0 >) (259) Now we consider anoter sared reservation sceme of cannels for -calls in te model wit a queue of o-calls Te main difference between tis sceme and

29 22 Models wit Queues for o-calls 59 te previous one is reallocation of -calls from te Secondary group to te Primary group Upon release of a cannel in te Primary group (eiter by an o-call or - call) an -call from te Secondary group is reallocated to te Primary group if and only if tere are no o-calls in te queue In oter words reallocation of -calls from te Secondary group to te Primary group is not allowed until te queue does not contain any o-calls Tis reservation sceme is called Handoff calls Overflow from Primary to Secondary wit Rearrangement (HOPSWR) Te system s state at any time is described by te two-dimensional vector k = (k, k 2 ), were k is te total number of busy cannels k = 0,,,m+n, and k 2 is te number of o-calls in te queue, k 2 = 0,,2, Te model s state space S as te following view: were S = n S i, S i Sj = Ø, i = j, (260) S 0 = {(j,0) : j = 0,,, m} {(m, j) : j =, 2, }, S i = {(m + i, j) : j = 0,, 2, }, i 0 On te basis of te adopted cannel allocation sceme, we can conclude, tat elements of te generating matrix q (k, k ), k, k S of appropriate 2-D MC are defined from te following relations (see Fig 23): λ o + λ if k m, k = k + e, λ if k m, k = k + e, λ o if k m, k q ( = k + e 2, k, k ) k = μ if k m, k = k e 2, mμ if k m, k = k e 2, k μ if k m, k 2 = 0, k = k e, (k m) μ if k m, k 2 0, k = k e, 0 in oter cases (26) Te desired QoS metrics of te system in tis case are defined via stationary distribution of te model as follows: P = p (m + n, i), (262) L o = ip (i), (263) were p (i) := k S p (k)δ(k 2, i) are marginal probability mass functions i=

30 60 2 Performance Analysis of Call-Handling Processes in Buffered CWN 00 λ λ λ λ 0 λ 0 λ m,0 m, m,2 µ 2µ mµ mµ mµ mµ λ (m+)µ µ µ λ 0 λ 0 λ 0 m+,0 m+, mµ m+,2 mµ mµ λ (m+n)µ nµ nµ λ 0 λ 0 λ 0 m+n,0 m+n, m+n,2 mµ mµ mµ Fig 23 State transition diagram for te HOPSWR model wit an unlimited queue of patient o-calls First, we will analyze te model of a macrocell wit an infinite queue for o-calls It is clear tat te following condition olds true in macrocells λ o >> λ Te above-mentioned condition on relations of intensities of different types of calls allow te conclusion tat transitions from state k S into state k + e 2 S occur more often tan into state k + e S In oter words transition between states (microstates) inside classes S i occurs more often tan transitions between states from different classes Because of tis fact, classes of microstates S i in (260) are depicted as isolated merged state <i>, and in initial state space S a known merging function is introduced Tus, an appropriate -D MC wit state space S := {< i >: i = 0,, 2,, n} is constructed Elements of te generating matrix of split models wit state space S i, denoted as q i (k, k ), k, k S i, are found troug relations (26): For te model wit state space S 0 : λ o + λ if k m, k = k + e, ( q 0 k, k ) λ o if k m, k = k + e 2, = k μ if k m, k = k e, mμ if k m, k = k e 2, 0 in oter cases; (264) For models wit state space S i,i : ( q i k, k ) λ o, if k = k + e 2, = mμ, if k = k e 2, 0 in oter cases (265)