An enhanced algorithm to solve multiserver retrial queueing systems with impatient customers

An Enhanced Algo i hm o Sol e Mul ise e Re ial Queueing Sys ems wi h Impa ien Cus ome s Tien Van Do a,∗Nam H. Do bJie Zhang c aMTA-BME In o ma ion Sys ems Resea ch G oup, Depa men o Ne wo ked Sys ems and Se ices, Budapes Uni e si y o Technology and Economics, H-1117, Magya ud´osok k¨o ´u ja 2., Budapes , Hunga y. bIn e -Uni e si y Cen e o Telecommunica ions and In o ma ics, Budapes Uni e si y o Technology and Economics, 4028 Deb ecen, Kassai ´u 26., Hunga y cCommunica ions G oup, he Depa men o Elec onic and Elec ical Enginee ing, Uni e si y o She ield, Mappin S ee , She ield, S1 3JD UK Abs ac The homogeniza ion o he s a e space o sol ing e ial queues e e s o an ap- p oach whe e he pe o mance o he M/M/c e ial queue wi h impa ien cus- ome s and cse e s is app oxima ed wi h a e ial queue wi h a maximum e ial a e es ic ed beyond a gi en numbe o use s in he o bi . As a consequence, he s a iona y dis ibu ion can be ob ained by he ma ix-geome ic me hod, which e- qui es he compu a ion o he a e ma ix. In his pape , we e isi an app oach based on he homogeniza ion o he s a e space. We p o ide he exac exp ession o he condi ional mean numbe o cus ome s based on he compu a ion o he a e ma ix Rwi h he ime complexi y o O(c). We de elop simpli ied equa ions o he memo y-e icien implemen a ion o he compu a ion o he pe o mance measu es. We cons uc an e icien algo i hm o he s a iona y dis ibu ion wi h he de e - mina ion o a h eshold ha allows he compu a ion o pe o mance measu es wi h a speci ic accu acy. Keywo ds: e ial queues, ma ix-geome ic me hod, spec al expansion, e icien algo i hm T. V. Do, N. H. Do, J. Zhang. An Enhanced Algo i hm o Sol e Mul ise e Re ial Queueing Sys ems wi h Impa ien Cus ome s. Compu e s & Indus ial En- ginee ing, DOI:10.1016/j.cie.2013.04.008, 2013 ∗Co esponding au ho Email add esses: do@hi .bme.hu (Tien Van Do), [email p o ec ed] (Nam H. Accep ed o Compu e s & Indus ial Enginee ing 1 In oduc ion Re ial queues ha e been used o ake in o accoun a phenomenon in mode n in o ma ion and elecommunica ion sys ems ha blocked cus ome s may e- eques o se ice a e a ce ain imeou [1–10]. In e ial queues a clien who does no ecei e he alloca ion o a se e joins he o bi and la e ini ia es a eques o se ice. The M/M/c e ial queue has been analyzed by many esea che s because he s a iona y dis ibu ion when he numbe o se e s is la ge han wo can be only ob ained using app oxima e echniques [1,6– 8,11,12]. Falin [13] p esen ed necessa y and su icien condi ions o e godici y o he e ial queues M/M/c. A well-known app oxima ion is based on he unca ion o he s a e space a a su icien ly la ge le el ela ed o he numbe o cus ome s in he o bi [13]. Ano he app oxima ion based on he homogeniza ion o he model was pionee ed by Neu s and Rao [14], whe e he M/M/c e ial queue is app oxima ed by he mul ise e e ial queue wi h he o al e ial a e ha does no depend on he numbe o clien s in he o bi as long as he o bi con ains he numbe o clien s g ea e han he speci ied alue N. No e ha he discussion o he choice o Nis p esen ed in he ecen book by A alejo and G´omez-Co al on e ial queues [8]. Wi h his assump ion, he s a iona y p obabili ies o he M/M/c e ial queue can be es ima ed by any algo i hm [15–19] based on he ma ix-geome ic me hod (MGM). Recen ly, Domenech-Benlloch e al. [20] conside ed a mul ise e e ial queue wi h he impa ien phenomenon o cus ome s wai ing in he o bi . They p o- posed wo di e en gene alized unca ed me hods (called HM1 and HM2) based on he homogeniza ion o he s a e space beyond a gi en numbe o use s in he e ial o bi . The s eady-s a e p obabili ies o he mul ise e e- ial queue wi h impa ien cus ome s a e app oxima ed wi h a modi ied e ial queue whe e he e ial a e beyond a ce ain le el only depends on he condi- ional mean alue o he numbe o cus ome s in he o bi . Domenech-Benlloch e al. [20] also compa ed hei me hods wi h o he well-known algo i hms ha belong o di e en ca ego ies [11] (app oxima ions, ini e unca ed me hods, gene alized unca ed me hods). The au ho s [20] showed ha he p oposed HM2 me hod ou pe o ms p e ious app oaches om he aspec o accu acy a he p ice o inc easing compu a ion cos . Based on he HM2 algo i hm o Domenech-Benlloch e al. [20], ou con ibu- ions allow an e icien compu a ion o he s a iona y dis ibu ion and he pe o mance measu es. Fi s , we e isi an app oach based on he homog- eniza ion o he s a e space and p o ide an e icien me hod wi h he ime complexi y o only O(c) o compu e he a e ma ix R. The me hod is based Do). 2 on a p ope y ha he cha ac e is ic ma ix polynomial has only a single non- ze o eigen alue and his single non-ze o eigen alue can be compu ed using he bisec ion me hod. Second, we de i e an exac exp ession o he condi ional mean numbe o cus ome s. Thi d, we de elop simpli ied equa ions ha allow he memo y-e icien implemen a ion o he compu a ion o he pe o mance measu es. Fou h, we cons uc an e icien compu a ion o he s a iona y dis- ibu ion wi h he de e mina ion o a h eshold, which gua an ees a speci ic accu acy o he compu a ion o pe o mance measu es. The es o his pape is o ganized as ollows. In Sec ion 2 we summa ize he conside ed queueing model wi h impa ien cus ome s. In Sec ion 3 we p esen ou new esul s ha se e as he ounda ions o he compu a ion. In Sec ion 4 we p o ide some nume ical esul s o illus a e he e iciency o ou algo i hm. Finally, Sec ion 5 concludes ou pape . 2 A Re ial Queueing Model wi h Impa ien Cus ome s We conside a e ial queueing model wi h chomogenous se e s and impa ien cus ome s. In e -a i al imes o cus ome s a e exponen ially dis ibu ed wi h pa ame e λ. Holding imes a e exponen ially dis ibu ed wi h pa ame e µ. Random a iable ג( ) ep esen s he numbe o occupied se e s a ime , hence 0 ≤ג( )≤cholds. A clien joins he o bi in o de o wai and e y upon when ג( ) = c. Le k( ) be he numbe o clien s in he o bi wai ing o e ial a ime . Each cus ome e ies wi h a e µ . Hence, he o al e ec i e e ial a e, when k( ) = j, is jµ . A e ying cus ome ei he lea es he queue wi h p obabili y Pim i all se e s a e busy upon he e ial o ejoins he o bi wi h p obabili y 1 −Pim. No e ha a ime be ween subsequen e ials o a speci ic use ollows he exponen ial dis ibu ion wi h pa ame e µ . This sys em can be ep esen ed by wo-dimensional con inuous- ime Ma ko chain (CTMC) Y={ג( ),k( )}wi h s a e space {0,1,...,c} × {0,1,...}. Le he s eady-s a e p obabili ies o CTMC Ybe deno ed by πi,j = lim →∞ P (ג( ) = i, k( ) = j). De ine he ow ec o j= [π0,j ,...,πc,j]. 2.1 No a ions CTMC Yis d i en by he ollowing ansi ions. (a) Aj(i, k) deno es he ansi ion a e om s a e (i, j) o s a e (k, j) (0 ≤ i, k ≤c;j= 0,1,...), which is caused by ei he he a i al o a cus ome (when i < c) o he lea ing o a clien a e he expi y o a holding ime. 3 Ma ix Ajis o size (c+ 1) ×(c+ 1) wi h elemen s Aj(i, k). Since Aj is j-independen , i can be w i en as Aj=A. The nonze o elemen s o Aja e Aj(i, i −1) = iµ o i= 1,...,c+ 1, and Aj(i, i + 1) = λ o i= 0,...,c. Because Ajis j-independen , i can be w i en as Aj=A=               0λ0... 0 0 0 µ0λ . . . 0 0 0 . . .. . .. . .. . .. . .. . .. . . 0 0 ... (c−1)µ0λ 0 0 ... 0cµ 0               ,∀j≥0. (b) Bj(i, k) ep esen s he one-s ep upwa d ansi ion a e om s a e (i, j) o s a e (k, j + 1) (0 ≤i, k ≤c;j= 0,1,...), which is caused by he a i al o a eques when all se e s a e busy (i.e., when i=c), hus inc easing k( ) by 1. Ma ix Bj(B, since i is j-independen ) is o size (c+ 1) ×(c+ 1) wi h elemen s Bj(i, k). The only nonze o elemen o Bj is Bj(c, c) = λ. Thus, we ge Bj=B=               000... 000 000... 000 . . .. . .. . .. . .. . .. . .. . . 0 0 ... 000 0 0 ... 0 0 λ               ,∀j≥0. (c) Cj(i, k) is he ansi ion a e om s a e (i, j) o s a e (k, j −1) (0 ≤ i, k ≤c;j= 1,2,...), which is due o he success ul e ial o a eques om he o bi . Ma ix Cjis o size (c+ 1) ×(c+ 1) wi h i s elemen s Cj(i, k). The nonze o elemen s o Cj(j≥1) a e Cj(i, i + 1) = jµ o i= 0,...,c and Cj(c, c) = jµ Pim. Ma ix Cj(∀j≥1) wi h elemen s Cj(i, k) is w i en as Cj=               0jµ 0... 000 0 0 jµ ... 000 . . .. . .. . .. . .. . .. . .. . . 0 0 ... 0 0 jµ 0 0 ... 0 0 jµ Pim               ,∀j≥1. 4 No e ha C0= 0 by de ini ion. Le DA,DCand DCj, j ≥1 deno e diagonal ma ices wi h he diagonal elemen s DA(i, i) = Pc k=0 A(i, k), DC(i, i) = Pc k=0 C(i, k) and DCj(i, i) = Pc k=0 Cj(i, k) o i= 0,...,c. The balance equa ions, which equa e he p oba- bili y luxes om and o he s a es o CTMC Y, and he no maliza ion equa- ion pe aining o CTMC Ycan be w i en as ollows (see [3,8]): 0Q(0) 1+ 1Q(1) 2=0,(1) j−1Q(j−1) 0+ jQ(j) 1+ j+1Q(j+1) 2=0(j≥1),(2) ∞ X j=0 jeT= 1.0 (no maliza ion), whe e Q(j) 0=B, j ≥0; Q(j) 1=A−DA−B−DCj, j ≥0; Q(j) 2=Cj, j ≥1 and eis he ow ec o o size c+ 1 wi h each elemen equal o uni y. Using he simila a gumen as in [3–5,8], he in ini esimal gene a o ma- ix [15,16,21] o Y, ha sa is ies [ 0, 1,...]QY=0, can be cons uc ed om equa ions (1) and (2) as ollows: QY=               Q(0) 1Q(0) 00 . . . . . . . . . . . . . . . . . . Q(1) 2Q(1) 1Q(1) 00 . . . . . . . . . . . . . . . 0Q(2) 2Q(2) 1Q(2) 00 . . . . . . . . . . . . 0 0 Q(3) 2Q(3) 1Q(3) 00 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q(j) 2Q(j) 1Q(j) 0. . . . . . . . . . . . . . . . . . Q(j+1) 2Q(j+1) 1Q(j+1) 0. . . . . . . . . . . . . . . . . . . . . Q(j+2) 2Q(j+2) 1Q(j+2) 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .               . I is clea ha QYis a block idiagonal ma ix wi h •QY(j, j + 1) = Q(j) 0, j ≥0,in he uppe diagonal, •QY(j, j) = Q(j) 1, j ≥0 in he main diagonal, •QY(j, j −1) = Q(j) 2, j ≥1 in he lowe diagonal. 2.2 An App oxima ion Domenech-Benlloch e al. [20] sugges ed ha he M/M/c e ial queue wi h impa ien cus ome s can be app oxima ed by he solu ion o he modi ied mul ise e e ial queue wi h he e ial a e µ (j) =      jµ i j < N M(N)µ i j≥N , 5 whe e M(N) = E[J|J≥N] is he condi ional mean numbe o cus ome s. As a consequence, he modi ied mul ise e e ial queue is desc ibed by a CTMC Z={גZ( ),kZ( )}wi h s a e space {0,1,...,c} × {0,1,...}, whe e גZ( ) ep esen s he numbe o occupied se e s a ime and kZ( ) is he numbe o clien s in he o bi wai ing o e ial a ime . The s eady-s a e p obabili ies o CTMC Za e deno ed by e πi,j = lim →∞ P (גZ( ) = i, kZ( ) = j), j≥0,0≤i≤c, and he ow ec o s e j= [e π0,j,...,e πc,j], j≥0. We de ine he ansi ion a e ma ices associa ed wi h CTMC Zas e Aj,e A,e Bj, e B,e Cjand e C o j≥0. No e ha we ha e e Aj=e A=Aand e Bj=e B=B o j≥0. Fu he mo e, e Cj=Cj o 0 ≤j < N and e Cj=e C=         0M(N)µ 0. . . 0 0 0 0 0 M(N)µ . . . 0 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 . . . 0 0 M(N)µ 0 0 . . . 0 0 M(N)µ Pim         , ∀j≥N. Fo j≥N, he balance equa ion o CTMC Zcan be ew i en as e j−1e Q0+e je Q1+e j+1 e Q2=0(j≥N),(3) whe e e Q0=e B, e Q1=e A−De A−e B−De C,e Q2=e C. The coe icien ma ices in he di e ence equa ions (3) a e j-independen . This leads o he ollowing solu ion based on he MGM (see [16]) e j=e N−1Rj−N+1 (j≥N−1),(4) whe e Ris he unique minimal nonnega i e solu ion o he quad a ic ma ix equa ion e Q0+Re Q1+R2e Q2= 0 (see [15,16]). A e he compu a ion o R, he a e ma ix, he s eady-s a e p obabili ies o s a es 0 ≤j≤N−1 can be de e mined by sol ing he balance equa ions pe aining o he le els 0≤j < N and he no maliza ion equa ion. 6 Algo i hm 1 The HM2 algo i hm M0(N) = N k= 0 epea k=k+ 1 Compu e Rma ix based on he loga i hmic educ ion algo i hm [16] Compu e Mk(N) using equa ion (5) un il |Mk(N)−Mk−1(N)|/Mk−1(N)< ǫM Sol e o j o j= 0,...,N Because Rand e Ndepend on M(N), we can ge he ixed-poin i e a ion M(N) = ∞ X j=N je je ∞ X j=Ne je =e N[R(I−R)−1+NI](I−R)−1e e N(I−R)−1e,(5) whe e Iis he iden i y ma ix o size (c+1)×(c+1). Hence, Domenech-Benlloch e al. p oposed Algo i hm 1 (called HM2) in [20]. 3 An Enhanced Algo i hm The s a iona y dis ibu ion o CTMC Yis app oxima ed by he s eady-s a e p obabili ies o CTMC Z. The e o e, we need o compu e he ollowing quan- i ies associa ed wi h CTMC Z: • he a e ma ix R, • he condi ional mean numbe o cus ome s M(N), • he s eady-s a e p obabili ies o s a es 0 ≤j≤N−1, • he es ima ion o N. No e ha Rcan be compu ed by he o iginal algo i hm o he MGM [15] and u he imp o ed algo i hms o MGM [16,18,19]. Howe e , he ime complexi y o hese algo i hms is O(c3). In wha ollows, we p o ide a me hod o compu e he a e ma ix (Theo em 1) in Sec ion 3.1. We de i e he exac and simpli ied o mula o he compu a ion o he condi ional mean numbe o cus ome s in he o bi (Co olla y 1). As a consequence, we can compu e he a e ma ix Rand he condi ional mean numbe M(N) o cus ome s in a e y e icien way. We p o ide a me hod o 7 de e mine he s eady-s a e p obabili ies o s a es 0 ≤j≤N−1 in Sec ion 3.2. We p o ide he new o mulae o pe o mance measu es and he ela ion be- ween pe o mance measu es in Sec ion 3.3. Nex , we p esen ou new esul and ou algo i hm o he compu a ion o he ini ial alue o Nin Sec ion 3.4. 3.1 The compu a ion o ma ix Rand M(N) In Theo em 1 we p o e ha he cha ac e is ic ma ix polynomial has only a single non-ze o eigen alue, and i can be compu ed using he bisec ion me hod. As a consequence, he a e ma ix Rhas a special o m and a me hod can be cons uc ed o compu e he a e ma ix Rwi h he compu a ional complexi y o O(c). The p ope y ha he cha ac e is ic ma ix polynomial has only a single non-ze o eigen alue allows he de i a ion o an exac equa ion o he condi ional mean numbe o cus ome s M(N). Theo em 1 The a e ma ix Rhas all ows o elemen s equal o ze o ex- cep he las ow = [ 0, 1,..., c], whe e c=xcis he single eigen alue o cha ac e is ic ma ix polynomial Q(x, M(N)) = e Q0+e Q1x+e Q2x2in he in e al (0,1) ( he co esponding le -eigen ec o is ψc= [ψc,0, ψc,2,...,ψc,c] wi h ψc,c = 1) and i=xcψc,i o 0≤i < c. The compu a ional complexi y o cand ψcis O(c). P oo . The s eady-s a e p obabili ies o he CTMC Za e exp essed as e j= c X k=0 bkψkxj−N+1 k(j≥N−1),(6) whe e bka e sui able coe icien s o be de e mined using he balance equa- ions pe aining o ows 0 o N−1 and he no maliza ion equa ion, (xk,ψk), k= 0,...,c a e he le eigen alue-eigen ec o pai s o Q(x, M(N)) = e Q0+e Q1x+e Q2x2inside he uni ci cle. They sa is y, ψkQ(xk, M(N)) = 0;de [Q(xk, M(N))] = 0, k = 0,...,c. Since he (c+ 1) ×(c+ 1) i-diagonal ma ix Q(x, M(N)) can be exp essed Q(x, M(N)) =        q1,1(x)q1,2(x) 0 . . . 0 0 q2,1(x)q2,2(x)q2,3(x). . . 0 0 0q3,2(x)q3,3(x). . . 0 0 . . . . . . . . . . . . . . . . . . 0 0 . . . qc,c−1(x)qc,c(x)qc,c+1(x) 0 0 . . . 0qc+1,c(x)qc+1,c+1(x)        , 8 whe e q1,1(x) = −(λ+M(N)µ )x, qi,i(x) = −(λ+M(N)µ + (i−1)µ)x (i= 2,...,c), qc+1,c+1(x) = λ−(λ+cµ +M(N)µ Pim)x +M(N)µ Pimx2, qi,i+1(x) = λx +M(N)µ x2(i= 1,...,c), qi+1,i(x) = µix (i= 1,...,c). I is easy o e i y ha Q(x, M(N)) has cze o-eigen alues. Le he null- eigen alues be x0, . . . , xc−1wi h co esponding independen le -eigen ec o s ψ0= [1,0,...,0], ψ2= [0,1,0,...,0],. . . ,ψc−1= [0,0,...,1,0], espec i ely. As a consequence, Q(x, M(N)) should ha e a single non-ze o eigen alue xc s ic ly inside he uni disk o ensu e ha he s a iona y dis ibu ion o CTMC e Yexis s. Le L(x, M(N)) and U(x, M(N)) deno e he componen ma ices in he LU decomposi ion o Q(x, M(N)) = L(x, M(N))U(x, M(N)) o any speci ic alue x. Due o he i-diagonal s uc u e, he componen ma ices o he LU decomposi ion o Q(x, M(N)) can be w i en as ollows L(x, M(N)) =       l1(x, M(N)) 0 0 ... 0 0 µx l2(x, M(N)) 0 ... 0 0 . . . . . . . . . . . . . . . . . . 0 0 . . . µ(c−1)x lc(x, M(N)) 0 0 0 . . . 0µcx lc+1(x, M(N))       , U(x, M(N)) =       1u1(x, M(N)) . . . 0 0 0 0 0 1 u2(x, M(N)) . . . 0 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 . . . 0 1 uc(x, M(N)) 0 0 . . . 0 0 1       . By equa ing he co esponding elemen s o Q(x, M(N)) and L(x, M(N)) · U(x, M(N)), and using some algeb aic simpli ica ions, we ge 9 P oposi ion 1 The nonse ice p obabili y is exp essed as Pns =λ−1Pimµ a2=λ−1Pimµ N−2 X m=1 me πc,m +(M(N)−1)e πc,N−1 1− c!.(26) P oo . Subs i u ing (16) o he de ini ion o a, we ob ain a=e NR(I−R)−1+NI(I−R)−1 =e NR(I+R 1− c ) + NI(I−R)−1 =e NR 1− c +NI(I−R)−1(using (13)) =e πc,N−1 R 1− c +NI(I−R)−1(using (14)) =e πc,N−1 c 1− c +N (I−R)−1(using (13)) =e πc,N−1M(N) (I−R)−1(using (12)) =M(N)e N(I−R)−1(using (13)) (27) =e πc,N−1M(N) I+R 1− c(using (16)) =e πc,N−1M(N) + c 1− c(using (14)) =e πc,N−1M(N) 1− c .(28) Subs i u ing (28) in o (25), we ob ain 16 a2= N−1 X m=0 me mz+a z = N−1 X m=0 me mz+e πc,N−1M(N) 1− c z = N−1 X m=1 me πc,m +e πc,N−1M(N) c 1− c = N−2 X m=1 me πc,m +e πc,N−1"N−1 + cM(N) 1− c# = N−2 X m=1 me πc,m +(M(N)−1)e πc,N−1 1− c (using (12)). (29) Equa ion (29) yields (26).✷ P oposi ion 2 The mean numbe o use s in he e ial o bi is N e =a1+a2.(30) P oo . F om he de ini ion o a1and a2, we ge a1+a2= N−1 X m=0 me m(o+z) + M(N)e N(I−R)−1(o+z) = N−1 X m=0 me me+M(N)e N(I−R)−1e. U ilizing (27) and he de ini ion o N e , we ob ain equa ion (30). ✷. P oposi ion 3 We can ob ain he blocking p obabili y Pbas ollows: Pb= N−2 X m=0 e πc,m +e πc,N−1 1− c .(31) P oo . U ilizing (16), we ob ain 17 Pb= N−1 X m=0 e mz+e N(I−R)−1z = N−1 X m=0 e mz+e N−1R(I+R 1− c )z = N−1 X m=0 e mz+e N−1 R 1− c z = N−1 X m=0 e mz+e πc,N−1 1− c z = N−2 X m=0 e mz+e N−1z+e πc,N−1 1− c z = N−2 X m=0 e πc,m +e πc,N−1+e πc,N−1 c 1− c = N−2 X m=0 e πc,m +e πc,N−1 1− c .✷. P oposi ion 4 The ollowing ela ion exis s be ween he pe o mance mea- su es N e =λ µ Pns(1 −Pim) Pim +Pb!.(32) P oo . F om Pds =λ−1µ a1,Pns =λ−1Pimµ a2,Pb=Pds +Pns and N e = a1+a2, we ge N e =λ µ (Pns Pim +Pds) =λ µ (Pns Pim +Pb−Pns) =λ µ Pns(1 −Pim) Pim +Pb!.✷ 3.4 An Es ima ion o N Domenech-Benlloch e al. [20] p esen ed some nume ical esul s conce ning choosing he app op ia e alue o h eshold N o achie e he equi ed accu acy o he app oxima ion o pe o mance measu es. Howe e , he au ho s [20] did no p esen a sys ema ic way o ind he app op ia e alue o h eshold N. In his sec ion, we will show an e icien me hod o es ima e h eshold N. 18 0.0 0.2 0.4 0.6 0.8 1.0 c 10 20 30 40 50 1 1- c -1 Fig. 1. 1/(1 − c)−1 s c 0 0.2 0.4 0.6 0.8 1 0 500 1000 1500 2000 2500 c N (c=50, ρ=0.8, Pim=0.2) µ =0.05 µ =0.1 µ =0.5 µ =1 µ =10 Fig. 2. c s N o c= 50, ρ= 0.8, Pim = 0.2 and µ= 1 F om equa ion (4), he ail o he dis ibu ion e jis geome ically dis ibu ed wi h pa ame e c. In he s able s a e o he sys em desc ibed by CTMC e Y, he highe he alue Nwe choose, he smalle he alue o Nis. I is an icipa ed ha he highe he chosen alue o Nis he smalle he alue o cis i he sys em is in he s able s a e (see Figu e 2). As one obse es om Figu e 1, he slope o he angen line o he cu e 1/(1 − c)−1 inc eases as capp oaches 1, and M(N)−N oo. Le h be he selec ed uppe limi o c( he pu pose o he selec ed uppe limi is o sea ch 19 o he ini ial alue o N o sa e he compu a ional ime), i.e., 0 < c≤ h. Algo i hm 3 To choose N 1: N←Nini 2: epea 3: N←N+ 1 4: un il lc+1( h, N −1 + 1/(1 − h)≤0 Algo i hm 4 P oposed algo i hm 1: Call algo i hm 3 o choose N 2: s ep ←1 3: epea 4: N←N+s ep 5: M0←N 6: k←0 7: epea 8: k←k+ 1 9: Compu e he oo xco lc+1(x, Mk−1) in he in e al (0, h] 10: Mk=N+ c 1− c(using equa ion (12)) 11: i e a ion e o =|Mk−Mk−1|/Mk−1 12: i lc+1(xc, Mk−1)> ǫ hen 13: k←0 14: N←N+ 1 15: M0←N 16: i e a ion e o = 1 17: end i 18: un il i e a ion e o < ǫM 19: Compu e ψcand R 20: Call Algo i hm 2 21: Compu e pe o mance measu es 22: Compu e Con e ge 23: un il Con e ge As s a ed be o e we also need o ind he ini ial alue o N. To sa e he com- pu a ional ime o he sea ch o he ini ial alue o N, we only check whe he lc+1(x, M(N)) has a oo in he in e al (0, h] ins ead o de e mining c. Be- cause lc+1(0, M(N)) = λholds, we ha e o examine whe he lc+1( h, M(N)) ≤ 0 in he i s s age o ou p oposed solu ion. Howe e , M(N) is no known in ad ance. To esol e his p oblem, Theo em 2 is applied o choose he ini ial alue o N(see Algo i hm 3), whe e lc+1( h, N −1+1/(1− h)) ≤0 is e i ied. Theo em 2 I cis he eigen alue o Q(x, M(N)) in he in e al (0,1), and lc+1( h, N −1 + 1/(1 − h)) ≤0 o h >0, hen cis bounded by h (i.e., 0< c≤ h). 20 P oo . Assume ha h < c, which ollows lc+1( h, M(N)) >0 because cis he eigen alue o Q(x, M(N)) in he in e al (0,1) (i.e., lc+1( c, M(N)) = 0) and lc+1(0, M(N)) = λ. No e ha lc+1(x, y) is a mono onically dec easing unc ion wi h espec o y o any cons an x, 0 < x < 1, because •lc+1(0, y) = λholds, •lc+1(x, y) has a single oo wi h espec o xin he in e al (0,1) o any cons an y, and • he highe he alue o yis he smalle he oo wi h espec o xis. We ha e N−1+1/(1− h)< M(N) = N−1+1/(1− c) om he assump ion. The e o e, lc+1( h, N −1 + 1/(1 − h)) > lc+1( h, M(N)). Using lc+1( h, M(N)) >0, we ob ain lc+1( h, N −1 + 1/(1 − h)) >0, which con adic s he gi en condi ion lc+1( h, N −1 + 1/(1 − h)) ≤0. The e o e, h < cdoes no hold, which yields ha cis bounded by h.✷ Theo em 2 is used in ou p oposed Algo i hm 4 o ind he ini ial alue o N. 3.5 The Con e gence C i e ion o a P oposed Algo i hm We p esen he de ails o a p oposed compu a ional p ocedu e in Algo i hm 4 ha in eg a es key esul s in Sec ion 3.1-3.4. In Algo i hm 4 wo loops a e applied a e he ini ial choice o N. The inne loop is needed o ind M(N) o a ce ain alue N, while he ou e loop is o une N o ob ain he equi ed accu acy o he es ima ion o pe o mance measu es. A no able ea u e o he p oposed p ocedu e compa ed o he HM2 algo i hm is ha he compu a ion o M(N) does no equi e he compu a ion o he s eady-s a e p obabili ies. Fu he mo e, he algo i hm is enhanced wi h he compu a ion o h eshold N. To de ine he con e gence c i e ion Con e ge we made he ollowing in es i- ga ion: we compu e he pe o mance measu es by unning he co e (be ween lines 4 and 19 o Algo i hm 3) o ou algo i hm wi h ixing N o pa ame e s µ= 1/180, µ = 0.01, Pim = 0.2, ǫM= 10−3and ǫ = 10−10 (Figu es 3 and 4). F om Figu es 3 and 4 he pe o mance measu es con e ges as Ng ows. Fu - he mo e, a high oscilla ion is obse ed in Figu e 4 as well (see he e-companion o mo e nume ical esul s wi h o he se ings o pa ame e s). The e o e, he con e gence c i e ion ( o de e mine when he pe o mance measu es each he “s able s a e”) is de ined as he ela i e e o be ween wo mo ing a e ages 21 conce ning a speci ic pe o mance measu e Con e ge := |PK i=Lχi/(K−L+ 1) −PK i=1 χi/K| PK i=Lχi/(K−L+ 1) < ǫp, whe e χi’s, i= 1,...,K, a e he la es alues o a chosen pe o mance measu e (e.g., N e ) de e mined so a by he algo i hm, ǫpis he speci ied accu acy, K and La e pa ame e s. I is ob ious ha he minimum choice o Kand Lis K= 3 and L= 2. Fu he mo e, he highe he alues o Kand La e, he mo e ime is needed, bu he be e he gua an ee o con e gence is ensu ed. In Table 1 we summa ize he compu a ional ime o he algo i hm o ρ= λ/(µc), µ= 1/180, µ = 0.01, Pim = 0.2. As obse ed he algo i hm success- ully s ops in he s able egion o pe o mance measu es (see Figu es 3 and 4). F om nume ical esul s (see he e-companion o esul s wi h o he se ings o Kand L), he minimum choice o K= 3 and L= 2 can gua an ee ha he p oposed algo i hm s eps o e he “oscilla ion pe iod”. Resul s in Table 1 and he e-companion empi ically show ha Pns can be se as he main pe - o mance measu e in he con e gence c i e ion Con e ge o de e mining all pe o mance measu es. 4 Compu a ional Times o he P oposed Algo i hm We plo he compu a ional ime e sus cand Nin Figu es 5 and 6, on a machine wi h In el Co eTM2 Duo T9400 2.53 GHz p ocesso (no e ha he algo i hm is implemen ed in Ma hema ica) o pa ame e s ρ=λ/(µc) = 0.8, µ= 1, Pim = 0.2, ǫ = 10−10,ǫM= 10−3. In he cu es he compu a ional ime o he HM2 algo i hm [20] and he co e (be ween lines 4 and 19 o Algo i hm 3) o ou algo i hm. As obse ed he compu a ional ime o he o iginal algo i hm HM2 is a apidly inc easing unc ion o c, while he compu a ional complexi y o he co e o ou algo i hm is only O(c). In Figu e 7, we plo he compu a ional ime o ou algo i hm e sus ρand c. I is obse ed om Table 1 and Figu e 7 ha ρimpac s he con e gence o he algo i hms. The explana ion behind his phenomenon is ha he highe ρis he mo e likely he oscilla ion is (see Figu e 4 and illus a ions in he e-companion). The e o e, mo e compu a ional ime is needed o s ep o e he oscilla ion. In o he wo ds, he algo i hm needs mo e ime o each he con- e gence due o he oscilla ion phenomenon a la ge alues o ρ. Howe e , he 22 9.27e-008 9.271e-008 9.272e-008 9.273e-008 9.274e-008 9.275e-008 9.276e-008 9.277e-008 9.278e-008 9.279e-008 9.28e-008 20 40 60 80 100 N e N 7.9e-009 7.92e-009 7.94e-009 7.96e-009 7.98e-009 8e-009 20 40 60 80 100 Pb N 7.84e-009 7.842e-009 7.844e-009 7.846e-009 7.848e-009 7.85e-009 7.852e-009 7.854e-009 7.856e-009 20 40 60 80 100 Pds N 9.6e-011 9.65e-011 9.7e-011 9.75e-011 9.8e-011 9.85e-011 9.9e-011 9.95e-011 1e-010 20 40 60 80 100 Pns N Fig. 3. Pe o mance measu es s N o c= 50, ρ= 0.4, µ= 1/180, µ = 0.01, Pim = 0.2 23 74.4 74.405 74.41 74.415 74.42 74.425 74.43 74.435 74.44 74.445 74.45 80 100 120 140 160 180 N e N 0.7521 0.75215 0.7522 0.75225 0.7523 0.75235 0.7524 0.75245 0.7525 80 100 120 140 160 180 Pb N 0.4618 0.46185 0.4619 0.46195 0.462 0.46205 0.4621 80 100 120 140 160 180 Pds N 0.2902 0.29025 0.2903 0.29035 0.2904 0.29045 0.2905 80 100 120 140 160 180 Pns N Fig. 4. Pe o mance measu es s N o c= 50, ρ= 1.4, µ= 1/180, µ = 0.01, Pim = 0.2 24 Table 1 Nand compu a ional ime o he algo i hm o K= 3, L = 2, ǫp= 10−5,ǫM= 10−3, ǫ = 10−10,ρ=λ/(µc), µ= 1/180, µ = 0.01, Pim = 0.2, h = 0.95 c= 50 c= 100 c= 200 c= 500 c= 1000 ρ N Time (s) NTime (s) NTime (s) NTime (s) NTime (s) 0.4 N e 17 0.889 6 0.765 8 1.435 39 16.318 13 13.121 Pb16 0.858 4 0.468 8 1.42 39 16.255 13 13.275 Pds 16 0.873 5 0.609 8 1.42 39 16.38 13 13.166 Pns 23 1.233 23 2.574 12 2.449 39 16.396 13 13.12 0.8 N e 38 2.511 29 3.697 46 9.703 35 15.646 13 15.287 Pb38 2.574 32 4.57 41 8.408 35 15.647 12 13.744 Pds 38 2.511 32 4.446 41 8.377 35 15.709 11 11.762 Pns 47 3.182 32 4.617 46 9.751 36 17.035 40 52.355 1.0 N e 44 2.855 76 7.504 117 14.133 206 35.491 363 82.619 Pb44 2.792 76 7.769 117 14.227 206 35.756 363 82.509 Pds 44 2.698 76 7.737 124 16.114 206 35.506 363 82.914 Pns 44 2.777 79 8.129 124 16.255 206 36.099 363 82.477 1.4 N e 102 3.042 198 7.957 397 26.411 858 81.417 1679 264.001 Pb100 2.621 198 8.064 397 26.427 858 81.589 1679 266.122 Pds 106 3.463 198 8.003 397 26.318 858 81.339 1679 264.781 Pns 106 3.464 202 9.375 397 26.458 858 81.354 1679 266.013 compu a ional ime complexi y is s ill o O(c) o a speci ic ρ. Rema k. I is wo h emphasizing ha app oaches belonging o he ca ego y “App oxima ions” (Domenech-Benlloch e al. [20]) p oduced unaccep able e - o s in mos cases. The e o e, we do no ocus on he compa ison wi h hese algo i hms in his pape . The HM2 algo i hm o e comes o he app oaches in he e m o he accu acy. Ou algo i hm has he same accu acy as he HM2. Howe e , i is much as e han he HM2 algo i hm. We ha e shown ha he compu a ional ime complexi y o ou algo i hm is o O(c). To ou bes knowledge, we do no know ha he e is any o he algo i hm which has he compu a ional ime complexi y o O(c) o he M/M/c e ial queue wi h im- pa ien cus ome s and has he same accu acy as o he HM2 algo i hm (see Domenech-Benlloch e al. [20] and Do [3] o he o e iew o he la es algo- i hms). 25 0.87 0.872 0.874 0.876 0.878 0.88 20 40 60 80 100 N e N 0.0331 0.0332 0.0333 0.0334 0.0335 0.0336 20 40 60 80 100 Pb N 0.0312 0.0314 0.0316 0.0318 0.032 0.0322 20 40 60 80 100 Pds N 0.0014 0.00145 0.0015 0.00155 0.0016 20 40 60 80 100 Pns N Fig. 2. Pe o mance measu es s N o c= 50, ρ= 0.8, µ= 1/180, µ = 0.01, Pim = 0.2 3 14.32 14.33 14.34 14.35 14.36 14.37 14.38 14.39 14.4 14.41 14.42 20 40 60 80 100 120 N e N 0.3324 0.3325 0.3326 0.3327 0.3328 0.3329 0.333 0.3331 0.3332 20 40 60 80 100 120 Pb N 0.286 0.2865 0.287 0.2875 0.288 20 40 60 80 100 120 Pds N 0.045 0.0455 0.046 0.0465 0.047 20 40 60 80 100 120 Pns N Fig. 3. Pe o mance measu es s N o c= 50, ρ= 1.0, µ= 1/180, µ = 0.01, Pim = 0.2 4 74.4 74.405 74.41 74.415 74.42 74.425 74.43 74.435 74.44 74.445 74.45 80 100 120 140 160 180 N e N 0.7521 0.75215 0.7522 0.75225 0.7523 0.75235 0.7524 0.75245 0.7525 80 100 120 140 160 180 Pb N 0.4618 0.46185 0.4619 0.46195 0.462 0.46205 0.4621 80 100 120 140 160 180 Pds N 0.2902 0.29025 0.2903 0.29035 0.2904 0.29045 0.2905 80 100 120 140 160 180 Pns N Fig. 4. Pe o mance measu es s N o c= 50, ρ= 1.4, µ= 1/180, µ = 0.01, Pim = 0.2 5 4.16183e-069 4.16183e-069 4.16184e-069 4.16184e-069 4.16185e-069 4.16185e-069 4.16186e-069 4.16186e-069 4.16187e-069 20 40 60 80 100 N e N 3.7275e-071 3.72755e-071 3.7276e-071 3.72765e-071 3.7277e-071 3.72775e-071 3.7278e-071 20 40 60 80 100 Pb N 3.723e-071 3.72305e-071 3.7231e-071 3.72315e-071 3.7232e-071 3.72325e-071 3.7233e-071 20 40 60 80 100 Pds N 4.48e-074 4.482e-074 4.484e-074 4.486e-074 4.488e-074 4.49e-074 4.492e-074 4.494e-074 4.496e-074 4.498e-074 4.5e-074 20 40 60 80 100 Pns N Fig. 5. Pe o mance measu es s N o c= 500, ρ= 0.4, µ= 1/180, µ = 0.01, Pim = 0.2 6 4.11e-005 4.111e-005 4.112e-005 4.113e-005 4.114e-005 4.115e-005 4.116e-005 4.117e-005 4.118e-005 4.119e-005 4.12e-005 20 40 60 80 100 N e N 1.81e-007 1.815e-007 1.82e-007 1.825e-007 1.83e-007 1.835e-007 1.84e-007 1.845e-007 1.85e-007 20 40 60 80 100 Pb N 1.81e-007 1.811e-007 1.812e-007 1.813e-007 1.814e-007 1.815e-007 1.816e-007 1.817e-007 1.818e-007 1.819e-007 1.82e-007 20 40 60 80 100 Pds N 7e-010 7.05e-010 7.1e-010 7.15e-010 7.2e-010 7.25e-010 7.3e-010 7.35e-010 7.4e-010 20 40 60 80 100 Pns N Fig. 6. Pe o mance measu es s N o c= 500, ρ= 0.8, µ= 1/180, µ = 0.01, Pim = 0.2 7 65.1 65.2 65.3 65.4 65.5 65.6 65.7 100 150 200 250 N e N 0.187 0.1875 0.188 0.1885 0.189 100 150 200 250 Pb N 0.1754 0.1756 0.1758 0.176 0.1762 0.1764 100 150 200 250 Pds N 0.0115 0.0116 0.0117 0.0118 0.0119 0.012 0.0121 0.0122 100 150 200 250 Pns N Fig. 7. Pe o mance measu es s N o c= 500, ρ= 1.0, µ= 1/180, µ = 0.01, Pim = 0.2 8 737 737.5 738 738.5 739 740 760 780 800 820 840 860 880 N e N 0.75325 0.753255 0.75326 0.753265 0.75327 0.753275 0.75328 0.753285 0.75329 740 760 780 800 820 840 860 880 Pb N 0.466 0.4665 0.467 0.4675 0.468 740 760 780 800 820 840 860 880 Pds N 0.285 0.2855 0.286 0.2865 0.287 740 760 780 800 820 840 860 880 Pns N Fig. 8. Pe o mance measu es s N o c= 500, ρ= 1.4, µ= 1/180, µ = 0.01, Pim = 0.2 9 Table 1 Nand he compu a ional ime o he p oposed algo i hm o K= 3, L= 2 ǫp= 10−5,ǫM= 10−3,ǫ = 10−10,ρ=λ/(µc), µ= 1/180, µ = 0.01, Pim = 0.2, h = 0.95 c= 50 c= 100 c= 200 c= 500 c= 1000 ρ N Time (s) NTime (s) NTime (s) NTime (s) NTime (s) 0.4 N e 17 0.889 6 0.765 8 1.435 39 16.318 13 13.121 Pb16 0.858 4 0.468 8 1.42 39 16.255 13 13.275 Pds 16 0.873 5 0.609 8 1.42 39 16.38 13 13.166 Pns 23 1.233 23 2.574 12 2.449 39 16.396 13 13.12 0.8 N e 38 2.511 29 3.697 46 9.703 35 15.646 13 15.287 Pb38 2.574 32 4.57 41 8.408 35 15.647 12 13.744 Pds 38 2.511 32 4.446 41 8.377 35 15.709 11 11.762 Pns 47 3.182 32 4.617 46 9.751 36 17.035 40 52.355 1.0 N e 44 2.855 76 7.504 117 14.133 206 35.491 363 82.619 Pb44 2.792 76 7.769 117 14.227 206 35.756 363 82.509 Pds 44 2.698 76 7.737 124 16.114 206 35.506 363 82.914 Pns 44 2.777 79 8.129 124 16.255 206 36.099 363 82.477 1.4 N e 102 3.042 198 7.957 397 26.411 858 81.417 1679 264.001 Pb100 2.621 198 8.064 397 26.427 858 81.589 1679 266.122 Pds 106 3.463 198 8.003 397 26.318 858 81.339 1679 264.781 Pns 106 3.464 202 9.375 397 26.458 858 81.354 1679 266.013 10 Table 2 Nand he compu a ional ime o he p oposed algo i hm o K= 4, L= 2, ǫp= 10−5,ǫM= 10−3,ǫ = 10−10,ρ=λ/(µc), µ= 1/180, µ = 0.01, Pim = 0.2, h = 0.95 c= 50 c= 100 c= 200 c= 500 c= 1000 ρ N Time (s) NTime (s) NTime (s) NTime (s) NTime (s) 0.4 N e 23 1.061 7 0.765 11 1.841 43 17.504 14 13.541 Pb17 0.78 5 0.484 11 1.809 43 17.487 14 13.588 Pds 17 0.749 5 0.499 11 1.779 43 17.254 14 13.462 Pns 23 1.06 36 3.213 14 2.559 43 17.316 14 13.447 0.8 N e 47 2.808 30 3.713 53 10.639 36 15.694 15 16.614 Pb47 2.761 34 4.352 46 8.705 36 15.631 12 12.465 Pds 47 2.761 34 4.337 46 8.814 36 15.616 12 12.387 Pns 47 2.793 34 4.305 53 10.608 37 16.926 43 52.057 1.0 N e 52 2.933 79 7.223 124 14.445 225 44.959 364 95.52 Pb52 2.932 79 7.332 124 14.617 225 44.85 364 95.02 Pds 52 2.933 79 7.363 124 14.633 225 44.975 364 94.584 Pns 61 3.698 83 7.971 138 18.362 225 44.788 364 94.833 1.4 N e 106 3.042 202 8.409 404 28.735 863 98.421 1684 332.251 Pb102 2.62 202 8.362 404 28.657 863 98.515 1684 331.315 Pds 106 3.011 202 8.377 404 29.343 863 98.405 1684 330.16 Pns 106 3.042 202 8.362 404 28.735 863 98.078 1684 332.641 11 Table 3 Nand he compu a ional ime o he p oposed algo i hm o K= 5, L= 2, ǫp= 10−5,ǫM= 10−3,ǫ = 10−10,ρ=λ/(µc), µ= 1/180, µ = 0.01, Pim = 0.2, h = 0.95 c= 50 c= 100 c= 200 c= 500 c= 1000 ρ N Time (s) NTime (s) NTime (s) NTime (s) NTime (s) 0.4 N e 29 1.389 14 1.295 12 2.122 55 23.447 15 15.351 Pb23 1.045 6 0.655 12 2.122 55 23.447 15 15.21 Pds 23 1.061 6 0.624 12 2.138 55 23.416 15 15.288 Pns 29 1.341 37 3.447 18 3.245 55 23.588 15 15.288 0.8 N e 49 2.932 32 3.993 56 11.715 37 17.316 16 18.642 Pb49 2.98 45 5.553 53 10.671 37 16.973 13 14.055 Pds 49 2.979 45 5.491 53 10.686 37 17.051 13 14.04 Pns 49 2.949 45 5.569 56 12.137 40 19.126 49 59.717 1.0 N e 54 3.198 83 7.94 138 18.346 229 52.573 386 134.051 Pb54 3.198 83 8.003 138 18.58 229 52.65 386 132.102 Pds 54 3.104 83 8.018 138 18.626 229 52.276 386 133.583 Pns 54 3.182 83 8.05 140 20.03 229 52.261 386 133.162 1.4 N e 111 3.573 204 9.547 407 32.947 872 123.319 1688 406.523 Pb106 3.01 204 9.454 407 32.76 872 123.287 1688 409.581 Pds 111 3.51 204 9.531 407 33.166 872 121.665 1688 411.032 Pns 111 3.573 204 9.563 407 32.838 872 122.321 1688 408.005 12