{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "# This file contain Maple code for the calculation of alpha given In Lemma 3 in the manus cript" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }} {PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "k:=8; # Number of servers - in this file we solve for 8 servers" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "rho:=lambda/(7*mu+mu1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rhoG*&%'lambdaG\"\"\",&*&\"\"(F'%#muGF'F' %$mu1GF'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "#The orde r of the variables in the vector: [p0,p10,p01,p11,p02,p12,p03,p13,p04, p14,p05,p15,p06,p16,p07]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "# p0= \+ steady stare probability the system has zero customers. p0j is the pro bability for zero at the heterogamous server and j in the homogenous s ervers. p1,j is similar where the heterogamous server is busy." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 895 "A := matrix( [[1,1,1,1,1,1, 1,1,1,1,1,1,1,1/(1-rho),1/(1-rho)],[-lambda,mu1,mu,0,0,0,0,0,0,0,0,0,0 ,0,0],[-lambda/8,lambda+mu1,0,-mu,0,0,0,0,0,0,0,0,0,0,0],[-7/8*lambda, 0,(lambda+mu),-mu1,-2*mu,0,0,0,0,0,0,0,0,0,0],[0,0,-6/7*lambda,0,lambd a+2*mu,-mu1,-3*mu,0,0,0,0,0,0,0,0],[0,0,0,0,-5/6*lambda,0,lambda+3*mu, -mu1,-4*mu,0,0,0,0,0,0],[0,0,0,0,0,0,-4/5*lambda,0,lambda+4*mu,-mu1,-5 *mu,0,0,0,0],[0,0,0,0,0,0,0,0,-3/4*lambda,0,lambda+5*mu,-mu1,-6*mu,0,0 ],[0,0,0,0,0,0,0,0,0,0,-2/3*lambda,0,lambda+6*mu,-mu1,-7*mu],[0,-lambd a,-lambda/7,lambda+mu1+mu,0,-2*mu,0,0,0,0,0,0,0,0,0],[0,0,0,-lambda,-l ambda/6,lambda+mu1+2*mu,0,-3*mu,0,0,0,0,0,0,0],[0,0,0,0,0,-lambda,-lam bda/5,lambda+mu1+3*mu,0,-4*mu,0,0,0,0,0],[0,0,0,0,0,0,0,-lambda,-lambd a/4,lambda+mu1+4*mu,0,-5*mu,0,0,0],[0,0,0,0,0,0,0,0,0,-lambda,-lambda/ 3,lambda+mu1+5*mu,0,-6*mu,0],[0,0,0,0,0,0,0,0,0,0,0,0,-lambda/2,-mu1*r ho,lambda+7*mu-mu1*rho]] );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"A GK%'matrixG6#7171\"\"\"F*F*F*F*F*F*F*F*F*F*F*F**&F*F*,&F*F**&%'lambdaG F*,&*&\"\"(F*%#muGF*F*%$mu1GF*!\"\"F4F4F+71,$F.F4F3F2\"\"!F7F7F7F7F7F7 F7F7F7F7F771,$*&\"\")F4F.F*F4,&F.F*F3F*F7,$F2F4F7F7F7F7F7F7F7F7F7F7F77 1,$*(F1F*F;F4F.F*F4F7,&F.F*F2F*,$F3F4,$*&\"\"#F*F2F*F4F7F7F7F7F7F7F7F7 F7F771F7F7,$*(\"\"'F*F1F4F.F*F4F7,&F.F**&FEF*F2F*F*FB,$*&\"\"$F*F2F*F4 F7F7F7F7F7F7F7F771F7F7F7F7,$*(\"\"&F*FIF4F.F*F4F7,&F.F**&FNF*F2F*F*FB, $*&\"\"%F*F2F*F4F7F7F7F7F7F771F7F7F7F7F7F7,$*(FWF*FRF4F.F*F4F7,&F.F**& FWF*F2F*F*FB,$*&FRF*F2F*F4F7F7F7F771F7F7F7F7F7F7F7F7,$*(FNF*FWF4F.F*F4 F7,&F.F**&FRF*F2F*F*FB,$*&FIF*F2F*F4F7F771F7F7F7F7F7F7F7F7F7F7,$*(FEF* FNF4F.F*F4F7,&F.F**&FIF*F2F*F*FB,$*&F1F*F2F*F471F7F6,$*&F1F4F.F*F4,(F. F*F3F*F2F*F7FCF7F7F7F7F7F7F7F7F771F7F7F7F6,$*&FIF4F.F*F4,(F.F*F3F**&FE F*F2F*F*F7FLF7F7F7F7F7F7F771F7F7F7F7F7F6,$*&FRF4F.F*F4,(F.F*F3F**&FNF* F2F*F*F7FUF7F7F7F7F771F7F7F7F7F7F7F7F6,$*&FWF4F.F*F4,(F.F*F3F**&FWF*F2 F*F*F7FgnF7F7F771F7F7F7F7F7F7F7F7F7F6,$*&FNF4F.F*F4,(F.F*F3F**&FRF*F2F *F*F7F^oF771F7F7F7F7F7F7F7F7F7F7F7F7,$*&FEF4F.F*F4,$*(F3F*F.F*F/F4F4,( F.F**&F1F*F2F*F*FcqF4Q(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "b := vector( [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bGK%'vectorG6#71\"\"\"\"\"!F*F*F*F *F*F*F*F*F*F*F*F*F*Q(pprint16\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "C:=linsolve(A, b ): #solve the linera system Ax=b" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "# recall that sequence of the vector: [p0,p10,p01,p11 ,p02,p12,p03,p13,p04,p14,p05,p15,p06,p16,p07]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "# pj is the probabiloty to have j costumers in t he system. Observe that p1=p01+p10, p4=p04+p13..." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 121 "p0:=C[1]: p1:=C[2]+C[3]: p2:=C[4]+C[5]: p3: =C[6]+C[7]: p4:=C[8]+C[9]: p5:=C[10]+C[11]: p6:=C[12]+C[13]: p7:=C[14] +C[15]: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "F:=simplify(p1 +p2*2+p3*3+p4*4+p5*5+p6*6+p7*(7-6*rho)/(1-rho)^2); #expected number of costumers in the system" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"FG**%' lambdaG\"\"\",hn**\"&S-$F')%#muG\"\"&F'F&F')%$mu1G\"\"$F'!\"\"**\"#%)F ')F&F-F'F,F'F.F'F1**\"$I'F')F&\"\"%F')F,\"\"#F'F.F'F1**\"%gLF')F&F0F') F,F0F'F.F'F1**\"&+E\"F')F&F:F')F,F8F'F.F'F1*&\"'gpCF')F,\"\"*F'F1*(\" \"(F')F&\"\"'F'F.F'F1*(\"#UF')F&FHF'F9F'F'*(\"&!GNF')F,FJF'F.F'F1*(FJF 'FMF')F/F:F'F'*(\"(!G*z\"F')F,\"\")F'F/F'F1*(\"'g*)\\F')F,FHF'FRF'F1*( \"%35F'F4F'FBF'F1**F3F'F4F'F>F'F/F'F1**\"$o\"F'FRF'F4F'F9F'F1**\"'+]JF 'FPF'FAF'F/F'F1**\"(!))=6F'FYF'F&F'F/F'F1**\"'!)GOF'FPF'F&F'FRF'F1**\" #@F'F/F'FIF'F9F'F'**\"%]JF'FRF'F7F'F>F'F1**\"'![B\"F'FAF'F+F'FRF'F1** \"&+/&F'F=F'F+F'F/F'F1**\"&+_#F'F=F'FBF'FRF'F1**\"%5WF'F7F'FBF'F/F'F1* *F`oF'FRF'FIF'F,F'F'*(\"&!))oF'F=F'FPF'F1*(\"'gVBF'FYF'FAF'F1*(\"&q>\" F'F7F'F+F'F1*(\"#xF'F>F'FIF'F'*(\"'gLUF'FUF'F&F'F1F',(*&FHF'F,F'F1F/F1 F&F'F1,N*(F3F'F>F'FIF'F'*(F*F'FYF'FAF'F'*&FMF'FRF'F'**\"#9F'F/F'FIF'F9 F'F'**F]qF'FRF'FIF'F,F'F'**\"$_#F'F4F'F>F'F/F'F'**\"$S)F'FRF'F7F'F>F'F '**\"%?DF'F7F'FBF'F/F'F'**\"$E\"F'FRF'F4F'F9F'F'**\"%+UF'F=F'FBF'FRF'F '**\"&?^\"F'FAF'F+F'FRF'F'**\"&+c(F'FPF'FAF'F/F'F'**\"&+o\"F'F=F'F+F'F /F'F'*(\"&?.%F'FYF'FRF'F'*(\"'SAGF'FUF'F/F'F'*(FHF'FMF'F9F'F'*(F6F'F4F 'FBF'F'*(F " 0 "" {MPLTEXT 1 0 104 "# The expected number of costumers in the system in the symmetric model where all servers are identical" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "Fsym:=simplify((lambda/mu)^8/8!*lambda/8 /mu/(1-lambda/8/mu)/(1-lambda/8/mu)/(sum((lambda/mu)^n/n!,n=0..7)+(lam bda/mu)^8/8!/(1-lambda/8/mu))+lambda/mu);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%FsymG,$*,\"\"#\"\"\",2*&\"'!Gh\"F()%#muG\"\"(F(!\"\" *(\"'g47F()F-\"\"'F(%'lambdaGF(F/*(\"&SG%F()F4F'F()F-\"\"&F(F/*(\"%S#* F()F4\"\"$F()F-\"\"%F(F/*(\"%g7F()F4F?F()F-F=F(F/*(\"#%)F()F4F9F()F-F' F(F/*(F.F()F4F3F(F-F(F(*&F=F()F4F.F(F(F(F4F(,2*&\"&?.%F(F,F(F(*(\"&!GN F(F2F(F4F(F(*(\"&?^\"F(F7F(F8F(F(*(\"%+UF(FF(F(*(\"$S)F(FBF(FCF(F (*(\"$E\"F(FFF(FGF(F(*(\"#9F(FIF(F-F(F(*$FKF(F(F/,&*&\"\")F(F-F(F/F4F( F/F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "simplify((subs(mu1 =mu,F)-Fsym)); # Verify that the the solution is identical to the symm etric case when substituting mu1=mu for the heterogamous server." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "evalf( subs(mu=5,lambda=28,Fsym)); #Calculating the expected number of consum ers in the symmetric model" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+UoSJ i!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "a:=simplify(subs(m u1=mu,diff(F,mu1,mu1))); #First component in alpha" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%\"aG,$*.\"\")!\"\"%'lambdaG\"\"\"%#muGF(,P*(\"-SYA1 >uF*)F+\"#7F*)F)\"#5F*F(*(\"3+!o6*f9LE9F*)F+\"#@F*F)F*F(*(\"0+78i!Gt>F *)F+\"#9F*)F)F'F*F(*(\"*Whde&F*)F+\"\"(F*)F)\"#:F*F(*(\"3+!3El6q(y9F*) F+\"#?F*)F)\"\"#F*F(*(\"2+?vUw:S#**F*)F+\"#>F*)F)\"\"$F*F(*(\"2+!)G%oG ;;[F*)F+\"#=F*)F)\"\"%F*F(*(\"1+#>/p4u;&F*)F+\"#;F*)F)\"\"'F*F(*(\"1+7 \\\\f^m6F*)F+FAF*)F)F?F*F(*(\"/+S]Xo+AF*)F+\"#8F*)F)\"\"*F*F(*(\"2+[/O K%=%y\"F*)F+\"#F*F**(\"&!GNF*FbpF*F)F*F**(\"&?^ \"F*FFF*F^pF*F**(\"%+UF*FLF*FfoF*F**(\"$S)F*FRF*FjoF*F**(\"$E\"F*FboF* FgqF*F**(F:F*FXF*F+F*F**$FgnF*F*FdrF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "alpha:=simplify(subs(mu1=mu,k/(k-1)*(a-1/k^2*diff(Fsy m,mu,mu)))/2); #calculating alpha" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#> %&alphaG,$*,\"\"#!\"\"%'lambdaG\"\"\",<*$)F)\"#7F*F**(\"#XF*)F)\"#6F*% #muGF*F**(\"$***F*)F)\"#5F*)F3F'F*F**(\"&!G9F*)F)\"\"*F*)F3\"\"$F*F**( \"'?Z9F*)F)\"\")F*)F3\"\"%F*F**(\"(S')3\"F*)F)\"\"(F*)F3\"\"&F*F**(\"( +'\\iF*)F3\"\"'F*)F)FNF*F**(\")g<%z#F*)F3FHF*)F)FJF*F**(\")+w(z*F*)F3F BF*)F)FDF*F**(\"*+/Rj#F*)F3FF*F**(\"*+CQ9&F*)F3F7F*)F)F'F*F**( \"*+S=`'F*)F3F2F*F)F*F**&\"*+cU1%F*)F3F.F*F*F*,2*&\"&?.%F*FRF*F**(\"&! GNF*FMF*F)F*F**(\"&?^\"F*FinF*FIF*F**(\"%+UF*FenF*FCF*F**(\"$S)F*FWF*F =F*F**(\"$E\"F*FSF*F8F*F**(\"#9F*FOF*F3F*F**$FGF*F*!\"#F3F(F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "evalf(subs(lambda=28,mu=5,al pha)); #value of alpha in the manuscript" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+PD>h$)!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{MARK "21 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }