5.7 Applications

This is quite a realistic model for a queue in which the numbers increase by 1 upon the arrival of a new customer and decrease by 1 when a customer departs after they have been served. Assuming that these occur randomly at respective rates $\mu$ and $\lambda$, irrespective of the queue size, except that no more customers join the queue if it is full (of size $n$), we can represent this by a cts MC model with TP rate matrix, illustrated for the case $n=4$:

The first and last rows represent reflecting barriers at 0 and $n$. The states all intercommunicate and we could determine the asymptotic distribution by solving $\pi Q=0$ for the invariant distribution.

However it is simpler to observe that the system satisfies detailed balance.

${\pi }_i{Q}_{i,i+1}={\pi }_{i+1}{Q}_{i+1,i}$ implies that

$${\pi }_i\mu ={\pi }_{i+1}\lambda \Rightarrow {\pi }_{i+1}=\frac{\mu }{\lambda }{\pi }_i(i=0,\mathrm{\dots },n-1).$$

Thus ${\pi }_j$ is a geometric progression and

$$\pi ={\pi }_0(1,\frac{\mu }{\lambda },{\left(\frac{\mu }{\lambda }\right)}^2,\mathrm{\dots },{\left(\frac{\mu }{\lambda }\right)}^n).$$

We divide the vector by its sum

$${\pi }_0\frac{1-{\left(\frac{\mu }{\lambda }\right)}^{n+1}}{1-\frac{\mu }{\lambda }}$$

to get

$$\pi =\frac{1-\frac{\mu }{\lambda }}{1-{\left(\frac{\mu }{\lambda }\right)}^{n+1}}(1,\frac{\mu }{\lambda },{\left(\frac{\mu }{\lambda }\right)}^2,\mathrm{\dots },{\left(\frac{\mu }{\lambda }\right)}^n).$$

A mobile phone “cell” can support a maximum number of phonecalls $N$; let new calls start with rate $\gamma$ and let each call finish with rate $\lambda$, so that if there are $i$ calls ongoing then the rate of calls finishing is $\lambda i$.

This is also known as the immigration/death model. Consider a system which can support a population (e.g. worms, bacteria, people) up to size $N$. Let immigrants arrive at the system with rate $\gamma$; any potential immigrants are turned away if there are $N$ immigrants in the system. Further suppose that immigrants die at a rate proportional to the number, $i$, of immigrants in the system $\lambda i$.

The rate matrix for this system has

$$Q_{i,i+1}=\gamma (i=0,\mathrm{\dots },N-1)\text{and}{Q}_{i,i-1}=\lambda i(i=1,\mathrm{\dots },N)\text{with}{Q}_{i,j}=0\text{for}|i-j|>1,$$

and so we may use detailed balance to show that the equilibrium distribution is given by

$${\pi }_i=\frac{{\left(\frac{\gamma }{\lambda }\right)}^i\frac{1}{i!}}{{\sum }_{j=0}^N{\left(\frac{\gamma }{\lambda }\right)}^j\frac{1}{j!}}.$$

(See coursework question for details).