4. Queueing Networks with loss and blocking-1

Model without Functions

Unlike the previous Mutex example, the equivalent functionless SAN model has the same number and size of automata and will therefore have the same product state space. In fact, to remove the functions of the previous model (QN1) we must introduce a large number of synchronized events that mostly represent possible (non-zero) evaluations of each function.

The function $f_0$ is replaced with several synchronized events to handle the two automata that represent the first service station. These events allow the arrival of a customer only if queue 0 is not full. It is necessary to include two synchronized events (one for each customer class) for each state in which an arrival is possible (states $0$ through $K_0 - 1$ ).

The function $f_3$ already appears in synchronized events (events $e_{13}$ and $e_{23}$ ) in QN1. These events synchronize the automaton from which a customer departs and the automaton to which the customer arrives. To remove the function, it is necessary to include into these synchronizations, a third automaton which is used to verify if queue 3 has an available slot into which the incoming customer can be placed. This can be achieved using a similar technique to that employed to remove function $f_0$ . It is necessary to replace each of the synchronized events $e_{13}$ and $e_{23}$ by as many synchronized events as the number of states of the automata for which an arrival is possible (i.e., $K_3 - 1$ ).

The functions $g_1$ and $g_2$ also appear in synchronized events in QN1. Each event synchronizes only two automata ( $\mathcal{A}^{(01)}$ and $\mathcal{A}^{(11)}$ for event $e_{01}$ and $\mathcal{A}^{(02)}$ and $\mathcal{A}^{(22)}$ for event $e_{02}$ ). However these functions refer to the automata describing the state of service center 3. In order to remove these functions it is necessary to extend the synchronized events $e_{01}$ and $e_{02}$ to include the corresponding automaton ( $\mathcal{A}^{(31)}$ for $e_{01}$ and $\mathcal{A}^{(32)}$ for $e_{02}$ ). Not only is the complexity of these synchronized events increased, but it is also necessary to split these events into different events with different rates according to the number of clients in service center 3.

Function $h_3$ is the most easily eliminated. It transforms local events consisting of the departure of class 2 customers in service center 3 (transitions from state $i$ to state $i-1$ state of $\mathcal{A}^{(32)}$ ) into events that are always synchronized with the first state of automaton $\mathcal{A}^{(31)}$ .