4.1 Queueing Networks with loss and blocking-1

Model with Functions

A SAN model equivalent to the queueing network model just presented may be defined with 6 automata and functional transition rates. Two automata are needed to describe each of the service centers visited by both classes of customer and one automaton to describes each of the service centers visited by only one class of customer. This SAN model is represented graphically in Figure  and we shall refer to it as QN1.

Figure :Queueuing Network Model -- QN1

Arrivals to and departures from the system are represented by local events since they affect only one automaton. The routing of customers between service centers occasions synchronized events since the state of two automata are altered simultaneously. We denote such events by $e_{ij}$ representing the departure of a customer from service center $i$ to service center $j$ . The event $e_{01}$ represents the departure of a class 1 customer from service center 0 to service center . The departure of class 2 customers from service center 0 (loss behavior) is also represented by a synchronized event even though it only changes the state of the automaton representing class 2 customers in service center 0 ($\mathcal{A}^{(02)}$ ). However since this only happens when queue 2 is full (automaton $\mathcal{A}^{(22)}$ is in its last state) a synchronized event that synchronizes the transition representing the departure of a customer from queue 0 (an arc from state $i$ to state $i-1$ in automaton $\mathcal{A}^{(02)}$ ) with the ``circular'' transition of the last state of automaton $\mathcal{A}^{(22)}$ .

Functional rates are used to represent:

• the capacity restriction of queues represented by automata 0 and 3
• the dependent service rates in service center 0

The function $f_0$ represents the capacity restriction in queue 0. It is evaluated as true (1) if there is room for another customer in queue 0, i.e., if the number of class 1 plus class 2 customers is less than the capacity of the queue. Hence both rates $\lambda_1$ and $\lambda_2$ must be multiplied by $f_0$ where

$$f_0 = \left( st \mathcal{A}^{(01)} + st \mathcal{A}^{(02)} \right) < K_0$$

Analogously, function $f_3$ represents the capacity restriction in queue 3, and the transition rates ($\mu_{11}$ and $\mu_{22}$ ) of the synchronized events $e_{13}$ and $e_{23}$ in automata $\mathcal{A}^{(11)}$ and $\mathcal{A}^{(22)}$ respectively must be multiplied by the function

$$f_3 = \left( st \mathcal{A}^{(31)} + st \mathcal{A}^{(32)} \right) < K_3$$

The dependent service rates in service center 0 are represented by two functions called respectively $g_1$ and $g_2$ . The function $g_1$ is inversely proportional to the number of class 1 customers in service center 1 (the state of automaton $\mathcal{A}^{(31)}$ ), i.e.:

$$g_1 = \frac{\mu_{01}}{(1 + st \mathcal{A}^{(31)})}$$

The service rate of class 2 customers is analogously represented by:

$$g_2 = \frac{\mu_{02}}{(1 + st \mathcal{A}^{(32)})}$$

The last function in this model represents the priority of class 1 over class 2 customers in service center 3. This function must be multiplied by the rate $\mu_{32}$ :

$$h_3 = st \mathcal{A}^{(31)} = 0$$

The product state space of this model is given by

$$pss = (K_0 + 1)^2 \times (K_1 + 1) \times (K_2 + 1) \times (K_3 + 1)^2$$

However, only some of these states are reachable since obviously, the sum of the states of automata representing the same service center cannot be greater than the capacity of the service center.

The textual .san  files describing this model are:

K1 =5, K2 =3, K3 =2, K4 =4, 

K1 =10, K2 =10, K3 =10, K4 =10,

Where : 

- Ki  is  the  capacity  of  queues.