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Table 6 The values of \((q^*_S, q^*_C, q^*_D)\) that maximize the weighted sum throughput \(T_w\) for different values of \(M_U\) when \(\alpha = 0.7\) and the queue at S is stable

From: A wireless caching helper system with heterogeneous traffic and random availability

\(\delta = 0.5\)
\(M_U\)   \(w = 1/4\) \(w = 2/4\) \(w = 3/4\)
200 max. \(T_w\) 0.227 0.285 0.342
\((q^*_S, q^*_C, q^*_D)\) (0.999, 1, 1) (0.999, 1, 1) (0.999, 1, 1)
400 max. \(T_w\) 0.224 0.283 0.341
\((q^*_S, q^*_C, q^*_D)\) (0.801, 1, 1) (0.824, 1, 1) (0.769, 1, 1)
600 max. \(T_w\) 0.221 0.281 0.340
\((q^*_S, q^*_C, q^*_D)\) (0.756, 1, 1) (1, 1, 1) (1, 1, 1)
800 max. \(T_w\) 0.219 0.279 0.340
\((q^*_S, q^*_C, q^*_D)\) (1, 1, 1) (1, 1, 1) (0.746, 1, 1)
1000 max. \(T_w\) 0.217 0.278 0.339
\((q^*_S, q^*_C, q^*_D)\) (0.732, 1, 1) (1, 1, 1) (1, 1, 1)
\(\delta = 1.2\)     
200 max. \(T_w\) 0.145 0.230 0.315
  \((q^*_S, q^*_C, q^*_D)\) (0.713, 1, 1) (0.713, 1, 1) (0.713, 1, 1)
400 max. \(T_w\) 0.135 0.223 0.312
  \((q^*_S, q^*_C, q^*_D)\) (1, 1, 0) (0.999, 1, 0) (0.999, 1, 0)
600 max. \(T_w\) 0.129 0.220 0.310
  \((q^*_S, q^*_C, q^*_D)\) (1, 1, 0) (1, 1, 0) (1, 1, 0)
800 max. \(T_w\) 0.126 0.217 0.309
  \((q^*_S, q^*_C, q^*_D)\) (1, 1, 0) (1, 1, 0) (1, 1, 0)
1000 max. \(T_w\) 0.123 0.216 0.308
  \((q^*_S, q^*_C, q^*_D)\) (1, 1, 0) (1, 1, 0) (0.540, 1, 0)