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qrf_noblo_mmi.m
1function [UN,QN,p2opt]=qrf_noblo_mmi(M,MR,K,N,mu,v,rt)
2%%% PARAMETERS %%%
3% f; % finite capacity queue
4% M, integer, > 0; % number of queues
5% MR, integer, > 0; % number of independent blocking configurations
6% BB {m in 1:MR, i in 1:M} >=0; % blocking state
7% K {i in 1:M}, integer, > 0; % number of phases for each queue
8% F {i in 1:M}, integer, > 0; % capacity
9% N, integer, >0; % population
10% mu {i in 1:M, k in 1:K(i), h in 1:K(i)} >=0; % completion transition rates
11% v {i in 1:M, k in 1:K(i), h in 1:K(i)} >=0; % background transition rates
12% r {i in 1:M, j in 1:M} >=0; % routing probabilities
13
14%%% VARIABLES %%%
15%var p2 {j = 1:M, nj = 1+(0:N), k = 1:K(j), i = 1:M, ni = 1+(0:N), h = 1:K(i), m = 1:MR} >= 0;
16%var e {i = 1:M, k = 1:K(i)} >=0;
17
18MR = 1;
19BB = zeros(1,M);
20F = repmat(N,M,1);
21
22q = zeros(M,M,max(K),max(K));
23for i = 1:M
24 for j = 1:M
25 for k = 1:K(i)
26 for h = 1:K(i)
27 if j ~= i
28 q(i,j,k,h) = rt(i,j)*mu(i,k,h);
29 else
30 q(i,j,k,h) = v(i,k,h)+rt(i,i)*mu(i,k,h);
31 end
32 end
33 end
34 end
35end
36
37x = [zeros(M*(N+1)*max(K)*M*(N+1)*max(K)*MR,1); zeros(M*max(K),1)];
38
39options = optimset('fmincon');
40options.Display = 'iter';
41%options.LargeScale = 'off';
42options.MaxIter = 100;
43%options.MaxFunEvals = 1e10;
44%options.MaxSQPIter = 500;
45%options.TolCon = 1e-8;
46%options.Algorithm = 'sqp';
47%options.OutputFcn = @outfun;
48
49[xopt, fopt] = fmincon(@(x) mmi(x),x,[],[],[],[],x*0,x*0+1,@(x) sub_qrfcon(x,q,M,MR,BB,F,N),options);
50[p2opt,~] = sub_qrfvar(xopt);
51
52for ti=1:M
53 UN(ti) = 0;
54 QN(ti) = 0;
55 for m=1:MR
56 for ni=1+(1:F(ti))
57 for ki=1:K(ti)
58 UN(ti) = UN(ti) + p2opt(ti,ni,ki,ti,ni,ki, m);
59 QN(ti) = QN(ti) + ni*p2opt(ti,ni,ki,ti,ni,ki, m);
60 end
61 end
62 end
63end
64
65 function fobj = mmi(x)
66 % MMI
67 %minimize MI: sum {m in 1..MR} sum {i in 1..M, j in 1..M, ki in 1..K[i], kj in 1..K[j]: i<>j} sum {nj in 0..F[j]} sum {ni in 0..F[j]} p2[i,ni,ki,j,nj,kj,m]*(log(1e-6+p2[i,ni,ki,j,nj,kj,m])-log(1e-6+p2[i,ni,ki,i,ni,ki,m])-log(1e-6+p2[j,nj,kj,j,nj,kj,m]));
68 [p2,~] = sub_qrfvar(x);
69 fobj = 0;
70 LOGTOL = 1e-6;
71 for m = 1:MR
72 for i = 1:M
73 for ki = 1:K(i)
74 for j = 1:M
75 if i~=j
76 for kj = 1:K(j)
77 for ni = 1+(1:F(i))
78 for nj = 1+(1:F(j))
79 fobj = fobj + p2(i,ni,ki,j,nj,kj,m)*(log(LOGTOL+p2(i,ni,ki,j,nj,kj,m))-log(LOGTOL+p2(i,ni,ki,i,ni,ki,m))-log(LOGTOL+p2(j,nj,kj,j,nj,kj,m)));
80 end
81 end
82 end
83 end
84 end
85 end
86 end
87 end
88 end
89
90 function [p2,e] = sub_qrfvar(x)
91 ctr = 1;
92 p2 = zeros(M,N+1,max(K),M,N+1,max(K),MR);
93 for j = 1:M
94 for nj = 1+(0:N)
95 for k = 1:K(j)
96 for i = 1:M
97 for ni = 1+(0:N)
98 for h = 1:K(i)
99 for m = 1:MR
100 p2(j,nj,k,i,ni,h,m) = x(ctr);
101 ctr = ctr + 1;
102 end
103 end
104 end
105 end
106 end
107 end
108 end
109 e = zeros(M,max(K));
110 for i=1:M
111 for k=1:K(i)
112 e(i,k) = x(ctr);
113 ctr = ctr + 1;
114 end
115 end
116 end
117
118 function [c,ceq] = sub_qrfcon(x,q,M,MR,BB,F,N)
119 c=sparse(0,1);
120 ceq=sparse(0,1);
121
122 %%% VARIABLES %%%
123 [p2,e] = sub_qrfvar(x);
124
125 %% DEFINITIONS
126 % subject to ONE {j in 1..M}: sum {nj in 0..N, k in 1..K[j], m in 1..MR} p2[j,nj,k,j,nj,k,m]=1;
127 for j = 1:M
128 % LHS
129 ceq(end+1) = 0;
130 for nj = 1+(0:N), for k = 1:K(j), for m = 1:MR
131 ceq(end) = ceq(end) + p2(j,nj,k,j,nj,k,m);
132 end, end, end
133 % RHS
134 ceq(end) = ceq(end) -1;
135 end
136
137 % subject to ZERO1 {j in 1..M, k in 1..K[j], nj in 0..N, i in 1..M, h in 1..K[i], ni in 0..N, m in 1..MR: i==j and nj==ni and h<>k}: p2[j,nj,k,i,ni,h,m]=0;
138 for j = 1:M, for k =1:K(j), for nj = 1+(0:N), for i = 1:M, for h = 1:K(i), for ni = 1+(0:N), for m = 1:MR
139 if i==j && (nj-1)==(ni-1) && h~=k % rescaled back nj and ni
140 ceq(end+1) = p2(j,nj,k,i,ni,h,m); %=0
141 end
142 end, end, end, end, end, end, end
143
144 % subject to ZERO2 {j in 1..M, k in 1..K[j], nj in 0..N, i in 1..M, h in 1..K[i], ni in 0..N, m in 1..MR: i==j and nj<>ni}: p2[j,nj,k,i,ni,h,m]=0;
145 for j = 1:M, for k =1:K(j), for nj = 1+(0:N), for i = 1:M, for h = 1:K(i), for ni = 1+(0:N), for m = 1:MR
146 if i==j && (nj-1)~=(ni-1) % rescaled back nj and ni
147 ceq(end+1) = p2(j,nj,k,i,ni,h,m); %=0
148 end
149 end, end, end, end, end, end, end
150
151 % subject to ZERO3 {j in 1..M, k in 1..K[j], nj in 0..N, i in 1..M, h in 1..K[i], ni in 0..N, m in 1..MR: i<>j and nj+ni>N}: p2[j,nj,k,i,ni,h,m]=0;
152 for j = 1:M, for k =1:K(j), for nj = 1+(0:N), for i = 1:M, for h = 1:K(i), for ni = 1+(0:N), for m = 1:MR
153 if i~=j && (nj-1)+(ni-1)>N % rescaled back nj and ni
154 ceq(end+1) = p2(j,nj,k,i,ni,h,m);
155 end
156 end, end, end, end, end, end, end
157
158 % subject to ZERO5 {j in 1..M, k in 1..K[j], i in 1..M, h in 1..K[i], ni in 0..F[i], m in 2..MR: BB[m,j]==1}: p2[j,0,k,i,ni,h,m]=0;
159 for j = 1:M, for k =1:K(j), for i = 1:M, for h = 1:K(i), for ni = 1+(0:F(i)), for m = 2:MR
160 if BB(m,j)==1
161 ceq(end+1) = p2(j,1+0,k,i,ni,h,m);
162 end
163 end, end, end, end, end, end
164
165 % subject to ZERO6 {j in 1..M, k in 1..K[j], nj in F[j]+1..N, i in 1..M, h in 1..K[i], ni in 0..N, m in 1..MR}: p2[j,nj,k,i,ni,h,m]=0;
166 for j = 1:M, for k =1:K(j), for nj = 1+((F(j)+1):N), for i = 1:M, for h = 1:K(i), for ni = 1+(0:N), for m = 1:MR
167 ceq(end+1) = p2(j,nj,k,i,ni,h,m);
168 end, end, end, end, end, end, end
169
170 % subject to ZERO7 {j in 1..M, k in 1..K[j], nj in 1..F[j], i in 1..M, h in 1..K[i], ni in 0..N, m in 2..MR: BB[m,j]==1 and i<>j and i<>f and ni+nj+F[f]>N}: p2[j,nj,k,i,ni,h,m]=0;
171 for j = 1:M, for k =1:K(j), for nj = 1+(1:F(j)), for i = 1:M, for h = 1:K(i), for ni = 1+(0:N), for m = 2:MR
172 if BB(m,j)==1 && i~=j && i~=f && (ni-1)+(nj-1)+F(f)>N % rescaled back ni and nj
173 ceq(end+1) = p2(j,nj,k,i,ni,h,m);
174 end
175 end, end, end, end, end, end, end
176
177 % subject to SIMMETRY {j in 1..M, nj in 0..N, k in 1..K[j], i in 1..M, ni in 0..N, h in 1..K[i], m in 1..MR}: p2[i,ni,h,j,nj,k,m] = p2[j,nj,k,i,ni,h,m];
178 for j = 1:M, for nj = 1+(0:N), for k =1:K(j), for i = 1:M, for ni = 1+(0:N), for h = 1:K(i), for m = 1:MR
179 ceq(end+1) = p2(i,ni,h,j,nj,k,m) - p2(j,nj,k,i,ni,h,m);
180 end, end, end, end, end, end, end
181
182 % subject to MARGINALS {j in 1..M, k in 1..K[j], nj in 0..N, i in 1..M, m in 1..MR: i<>j}: p2[j,nj,k,j,nj,k,m]= sum {ni in 0..N-nj} sum {h in 1..K[i]} p2[j,nj,k,i,ni,h,m];
183 for j = 1:M, for k =1:K(j), for nj = 1+(0:N), for i = 1:M, for m = 1:MR
184 if i~=j
185 % LHS
186 ceq(end+1) = p2(j,nj,k,j,nj,k,m);
187 % RHS
188 for ni = 1+(0:(N-nj)) %% added +1
189 for h = 1:K(i)
190 ceq(end) = ceq(end) - p2(j,nj,k,i,ni,h,m);
191 end
192 end
193 end
194 end, end, end, end, end
195
196 %subject to UEFF {j in 1..M, i in 1..M, ki in 1..K[i]}: e[i,ki] = sum {nj in 0..N, kj in 1..K[j], m in 1..MR, ni in 1..N: BB[m,i]==0} p2[j,nj,kj,i,ni,ki,m];
197 for j = 1:M, for i = 1:M, for ki = 1:K(i)
198 % LHS
199 ceq(end+1) = e(i,ki);
200 % RHS
201 for nj = 1+(0:N), for kj = 1:K(j), for m = 1:MR, for ni = 1+(1:N)
202 if BB(m,i)==0
203 ceq(end) = ceq(end) - p2(j,nj,kj,i,ni,ki,m);
204 end
205 end, end, end, end
206 end, end, end
207
208 %subject to THM1 {i in 1..M, k in 1..K[i]}: sum {j in 1..M, h in 1..K[i]} q[i,j,k,h]*e[i,k] =sum {j in 1..M, h in 1..K[i]} q[i,j,h,k]*e[i,h];
209 for i = 1:M, for k =1:K(i)
210 ceq(end+1) = 0;
211 % LHS
212 for j = 1:M, for h = 1:K(i)
213 ceq(end) = ceq(end) + q(i,j,k,h)*e(i,k);
214 end, end
215 % RHS
216 for j = 1:M, for h = 1:K(i)
217 ceq(end) = ceq(end) - q(i,j,h,k)*e(i,h);
218 end, end
219 end, end
220
221 %subject to THM2 {j in 1..M, k in 1..K[j], nj in 0..F[j], m in 1..MR}: sum {i in 1..M, ni in 1..F[i], ki in 1..K[i]} ni*p2[j,nj,k,i,ni,ki,m]= N*p2[j,nj,k,j,nj,k,m];
222 for j = 1:M, for k =1:K(j), for nj = 1+(0:F(j)), for m = 1:MR
223 ceq(end+1) = 0;
224 % LHS
225 for i = 1:M, for ni = 1+(1:F(i)), for ki = 1:K(i)
226 ceq(end) = ceq(end) + (ni-1)*p2(j,nj,k,i,ni,ki,m); % recaled back ni
227 end, end, end
228 % RHS
229 ceq(end) = ceq(end) - N*p2(j,nj,k,j,nj,k,m);
230 end, end, end, end
231
232 %subject to COR1 : sum {m in 1..MR, i in 1..M, j in 1..M, nj in 1..F[j], ni in 1..F[i], ki in 1..K[i], kj in 1..K[j]} ni*nj*p2[j,nj,kj,i,ni,ki,m]= N^2;
233 ceq(end+1) = 0;
234 % LHS
235 for m = 1:MR, for i = 1:M, for j = 1:M, for nj = 1+(1:F(j)), for ni = 1+(1:F(i)), for ki = 1:K(i), for kj = 1:K(j)
236 ceq(end) = ceq(end) + (ni-1)*(nj-1)*p2(j,nj,kj,i,ni,ki,m); % rescaled back ni and nj
237 end, end, end, end, end, end, end
238 % RHS
239 ceq(end) = ceq(end) - N^2;
240
241 % subject to THM4 {j in 1..M, k in 1..K[j], i in 1..M, m in 1..MR}: sum{t in 1..M} sum {h in 1..K[t]} sum {nj in 0..N} sum {nt in 0..N} nt*p2[j,nj,k,t,nt,h,m]
242 % >= N*sum {h in 1..K[i]} sum {nj in 0..N} sum {ni in 1..N} (p2[j,nj,k,i,ni,h,m]);
243 for j = 1:M, for k = 1:K(j), for i = 1:M, for m = 1:MR
244 c(end+1) = 0; % <= inequality
245 % LHS with sign swapped since >= in GLPK
246 for t = 1:M, for h = 1:K(t), for nj = 1+(0:N), for nt = 1+(0:N)
247 c(end) = c(end) - (nt-1)*p2(j,nj,k,t,nt,h,m); % rescaled back nt
248 end, end, end, end
249 % RHS with sign swapped since >= in GLPK
250 for h = 1:K(i), for nj = 1+(0:N), for ni = 1+(1:N)
251 c(end) = c(end) + N*(p2(j,nj,k,i,ni,h,m));
252 end, end, end
253 end, end, end, end
254 end
255end
256
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