Limited Recursion and Vanishing States

In this case recursion is also used to generate the bulks of messages, but it has an fixed limit defined by the positive integer constant $N$, which along with the constant probability $P$, form the set of parameters for the binomial distribution.

Figure: The Binomial_Bulk object.

As figure shows, whenever the Binomial_Bulk object receives a message, it can be forwarded to the server if the action with probability $P$ is executed, and then, in either one of the action codes, it will check whether the state variable $TempN$ is greater than zero. In the positive case, it will decrease $TempN$ by one and send another message to itself. Otherwise it will reset $TempN$ to $N$ and finish the recursion.

It is clear from the previous explanation and from the code, that the recursion in the Binomial_Bulk object takes place at most $N$ consecutive times. It can happen less than $N$ times if the process reaches a state where the queue is full.

The important thing to notice in this example is that, in order to generate a fixed size recursion, we need to add a new state variable to the model for keeping track of how deep the recursion is so far.

In the previous example, the model did not require this, because it refered to a geometric distribution, which has the memoryless property. This way, each step of the recursion can be determined without the use of any memory (state variables).

Since the geometric distribution is also the only discrete memoryless distribution, any other kind of batch size distribution (including the binomial case) will have to add new state variables to allow Tangram-II to generate the model's state space and transition rates.

At first, one might think this is bad, since adding a new state variable might increase the state space, but it turns out this does not happen. To explain why, we need to consider an example of what happens during the chain generation procedure for this model.

Suppose the model is set initially to the state where the queue is empty, and let our state tuple be defined as $(Server\_Queue.Queue, Binomial\_Bulk.TempN)$. Since the $Binomial\_Bulk$ object requires that $TempN$ be initialized to $N$, and having $N=3$ in this example, our initial state will be $(0,3)$. We further assume that the maximum number of customers in the queueing system is 10.

When an arrival event happens, while the model is in the initial state, the sequence of execution in the model can be illustrated by the diagram shown in figure .

Figure: The Binomial bulk generation process from the initial state.

Notice that the recursion ends when $TempN$ reaches zero, being reset to $N=3$. All the message exchanging happens in zero time, with the generation of several intermediate states, in which the value of $TempN$ can be 0, 1 and 2, but after the recursion the model always ends up with $TempN=3$. Because of this, all the intermediate states do not show up in the final state space (as shown in figure ), and so are called .

Figure: Transitions generated by the arrival event at the initial state.

After generating the state space for this model, all the states will have $TempN=N$. The main lesson to learn here is that even though the $TempN$ variable is not necessary to describe the state space (since it equals the constant $N$ in all states), it is essential to the state generation procedure.