**[1]** In fact a major concern with queues is that their length does not explode. This issue cannot be properly explored unless the state space is allowed to be infinite.\n",
"\n",
"

**[2]** An excellent introduction to operator semigroups, combined with\n",
"applications to PDEs and Markov processes, can be found in [[Applebaum, 2019](https://jstac.github.io/continuous_time_mcs/zreferences.html#id5)].\n",
"\n",
"

**[3]** Convergence of the sum in [(6.3)](#equation-opexpo) follows from boundedness of $ A $ and the fact that $ \\linop $ is a Banach space.\n",
"\n",
"

**[4]** Be careful: the definition of a UC semigroup requires that\n",
"$ t \\mapsto U_t $ is continuous as a map into $ \\linop $, rather than uniformly\n",
"continuous. The UC terminology comes about because, for a UC semigroup, we\n",
"have, by definition of the operator norm,\n",
"$ \\sup_{\\| g \\| \\leq 1} \\| U_s g - U_t g \\| \\to 0 $ when $ s \\to t $.\n",
"\n",
"

**[5]** An excellent treatment of the general theory of $ C_0 $ semigroups, can be found in [[Bobrowski, 2005](https://jstac.github.io/continuous_time_mcs/zreferences.html#id7)]."
]
}
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