Shoe Shine Shop

Example

Consider a shoe shine establishment consisting of two chairs — chair 1 and chair 2. A customer upon arrival goes initially to chair 1 where shoes are cleaned and polish is applied. After this, the customer moves to chair 2 where the polish is buffed.

Assumptions:

• Service times at chair 1 are $\text{Exp}({\mu }_1)$
• Service times at chair 2 are $\text{Exp}({\mu }_2)$
• Potential customers arrive according to a Poisson process with rate $\lambda$
• A customer will enter only if both chairs are empty

State Space

Define $X(t)$ as the current chair a customer is in. The state space is $S=\{0,1,2\}$:

State	Interpretation
0	System is empty
1	A customer is in chair 1
2	A customer is in chair 2

CTMC Parameters

Transition rates (Q-matrix):

$Q=\left(\begin{array}{ccc}-\lambda & \lambda & 0\\ 0& -{\mu }_1& {\mu }_1\\ {\mu }_2& 0& -{\mu }_2\end{array}\right)$

Sojourn rates: ${\nu }_0=\lambda$, ${\nu }_1={\mu }_1$, ${\nu }_2={\mu }_2$

Transition probabilities:

$P=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right)$

That is: $P_{01}=P_{12}=P_{20}=1$, and all other $P_{ij}=0$.

Related

• Continuous-Time Markov Chain
• Transition Rate Matrix