Properties of Markovian arrival processes

Below, we summarize some of the basic properties of the MAP. First, the set of MAPs is quite broad and, in theory, any stationary point process can be approximated arbitrarily closely by a MAP.

Proposition 6   [12] The set of MAPs is dense in the set of all the stationary point processes.

Second, the set of MAPs is closed under some operations. In particular, a superposition of two independent MAPs is a MAP.

Proposition 7   [117] A superposition of two independent MAPs, MAP( $\mathbf{C_0},\mathbf{C_1}$) and MAP( $\mathbf{D_0},\mathbf{D_1}$) is a MAP, MAP( $\mathbf{E_0},\mathbf{E_1}$), where

$$\mathbf{E_0} = \mathbf{C_0}\otimes \mathbf{D_0} \quad\mbox{and}\quad \mathbf{E_1} = \mathbf{C_1}\otimes \mathbf{D_1}.$$

Here, $\otimes$ denotes the Kronecker product.

Third, the average rate of events and the marginal distribution of the inter-event time of a MAP have simple mathematical expressions.

Proposition 8   [117,92] The average rate of events (the number of events in a unit time) in a MAP is called the fundamental rate of the MAP. The fundamental rate of MAP( $\mathbf{D_0},\mathbf{D_1}$) is given by

$$\lambda = \Vec{\theta} \mathbf{D_1} \Vec{1},$$

where $\Vec{\theta}$ is the stationary probability vector in the Markov chain with infinitesimal generator $\mathbf{D}=\mathbf{D_0}+\mathbf{D_1}$ (i.e., $\Vec\theta\mathbf{D}=\mathbf{0}$ and $\Vec\theta \Vec{1} = 1$).

The marginal distribution of the inter-event time of the above MAP is a PH( $\Vec{\phi},\mathbf{D_0}$) distribution, where

$$\Vec{\phi} = \frac{\Vec\theta \mathbf{D_1}}{\lambda},$$

which is the stationary probability vector immediately after the event. Note the difference between PH( $\Vec{\phi},\mathbf{D_0}$) and PH( $\Vec{\theta},\mathbf{D_0}$). PH( $\Vec{\theta},\mathbf{D_0}$) is the distribution of the time from an arbitrary epoch, i.e., the excess of the inter-event time.

Since the marginal distribution of the inter-event time in a MAP is a PH distribution, its moments, density function, and distribution function can be calculated via Proposition 4.

The variability of a point process can be characterized by the covariance of the inter-event times or the index of dispersion for intervals. The index of dispersion for intervals, IDI($\cdot$), of a MAP is defined by

$${\rm IDI}(i) = \frac{Var(S_i)}{i/\lambda^2},$$

where $S_i$ is the sum of the first $i$ inter-event times of the MAP. Observe that IDI($i$) is the ratio of the variance of $S_i$ to the corresponding variance of a Poisson process with the same rate. The MAP have a convenient mathematical expressions for the covariance and the index of dispersion for intervals.

Proposition 9   [7,92] Consider MAP( $\mathbf{D_0},\mathbf{D_1}$) with the fundamental rate $\lambda$. Recall the vectors, $\Vec\theta$ and $\Vec\phi$, of the MAP defined in Proposition 8. The covariance between the two inter-event times separated by $i-1$ events of the MAP is given by

$$\lambda^{-1} \Vec\theta \left( (-\mathbf{D_0}^{-1} \mathbf{D_1})^i - \Vec{1}\Vec{\phi}\right) (-\mathbf{D_0})^{-1} \Vec{1}.$$

The IDI($\cdot$) of the above MAP is given by

$$\begin{eqnarray*} IDI(i) & = & 2 \lambda \Vec\theta \left(\mathbf{I} + \mathbf{... ...} (-\mathbf{D_0}^{-1}\mathbf{D_1}) (-\mathbf{D_0})^{-1} \Vec{1}. \end{eqnarray*}$$

Finally, a MAP can be translated into a counting process, i.e., the number of events by time $t$, $N_t$. The MAP has a simple $z$-transform of $N_t$, and this leads to a convenient mathematical expression for the index of dispersion for counts, which characterizes the variability of a MAP via the variability of $N_t$. The index of dispersion for counts, IDC($\cdot$), of the MAP is defined by

$${\rm IDC}(t) = \frac{Var(N_t)}{\mbox{{\bf\sf E}}\left[ N_t \right]}.$$

Observe that IDC($t$) is the ratio of the variance of $N_t$ to the corresponding variance of a Poisson process with the same rate.

Proposition 10   [7] Let $N_t$ be the number of events by time $t$ in MAP( $\mathbf{D_0},\mathbf{D_1}$). Recall vectors, $\Vec\theta$ and $\Vec\phi$, and the fundamental rate $\lambda$ of the MAP defined in Proposition 8. Let $P(i,t)$ be the probability that $N_t = i$. The $z$-transform of $P(\cdot,t)$, $\hat P(z,t)\equiv \sum_{i=0}^\infty z^i P(i,t)$, is given by

$$\hat P(z,t) = \exp((\mathbf{D_0}+z\mathbf{D_1})t).$$

for $\vert z\vert\leq 1$, $t\geq 0$.

The IDC($\cdot$) of the above MAP is given by

$${\rm IDC}(t) = 1 - 2\lambda + \frac{2}{\lambda}\Vec\theta\ma... ...t} \Vec{c} \left( \mathbf{I} - e^{\mathbf{D}t}\right) \Vec{d},$$

where

$$\Vec{c} = \Vec\theta \mathbf{D_1} (\Vec{1}\Vec\theta - \math... ...} = (\Vec{1}\Vec\theta - \mathbf{D})^{-1}\mathbf{D_1} \Vec{1}.$$