Cluster Modeling of Lindley Process with Application to Queuing

Abstract

We investigate clusters of extremes defined as subsequent exceedances of high thresholds in a Lindley process. The latter is usually used to model the waiting time or the length of a queue in queuing systems. Distributions of the cluster and inter-cluster sizes of the Lindley process are obtained for a given value of the threshold assuming that the process begins from the zero value. An example of a M/M/1 queue and the impact of service and arrival rates on the cluster and inter-cluster distributions are shown.

Appendix

Proof

The expression for $E(T_2(x_{\rho }))$ (formula (15)) follows from the application of the generating function method. Let us denote by $\mathcal {C}(z)$ — the probability generating function of Catalan numbers $C_j$, i.e. $\mathcal {C}(z)=\sum _{j=0}^\infty C_j z^j$, $0<z<1/4$. It is known that $\mathcal {C}(z)={1-\sqrt{1-4z}\over 2z}$. Since $\mathcal {C}(z)$ satisfies the functional equation $\mathcal {C}(z)=1+z \mathcal {C}(z)^2$, the derivative of $\mathcal {C}(z)$ is equal to $\mathcal {C}'(z)={\mathcal {C}(z)^2 \over 1 - 2 z \mathcal {C}(z)}$. Thus the value of $E(T_2(x_\rho ))$ is equal to

$$\begin{aligned} E(T_2(x_\rho ))= & {} \sum _{j=2}^\infty j P\{T_2(x_\rho )=j\} = \sum _{j=2}^\infty j C_{j-1} \frac{\rho ^{j-3}(1-\rho ^2)}{(1+\rho )^{2j}}\left( 1- \rho \ln \left( {1-\rho ^2\over \rho }\right) \right) \\= & {} \sum _{j=1}^\infty (j+1) C_{j} \frac{\rho ^{j-2}(1-\rho ^2)}{(1+\rho )^{2(j+1)}}\left( 1- \rho \ln \left( {1-\rho ^2\over \rho }\right) \right) \\= & {} \frac{(1-\rho ^2)}{\rho ^2(1+\rho )^{2}} \left( 1- \rho \ln \left( {1-\rho ^2\over \rho }\right) \right) \\\times & {} \left( \underbrace{ \sum _{j=1}^\infty C_{j} [{\rho \over (1+\rho )^2}]^{j} }_{=\mathcal {C}\left( \frac{\rho }{(1+\rho )^{2}} \right) -1} + {\rho \over (1+\rho )^2} \underbrace{\sum _{j=1}^\infty jC_{j} [{\rho \over (1+\rho )^2}]^{j-1} }_{=\mathcal {C}'\left( \frac{\rho }{(1+\rho )^{2}} \right) } \right) . \end{aligned}$$

Now we have

$$\begin{aligned} \mathcal {C} \left( \frac{\rho }{(1+\rho )^{2}} \right) -1= & {} {1-\sqrt{1-4\frac{\rho }{(1+\rho )^{2}}}\over 2\frac{\rho }{(1+\rho )^{2}}}-1 = {1- {1 \over 1+\rho } \sqrt{1 + 2\rho + \rho ^2 - 4 \rho } \over 2\frac{\rho }{(1+\rho )^{2}}}-1 \\= & {} {(1+\rho )^{2}- (1+\rho ) \sqrt{(1 - \rho )^2} \over 2 \rho }-1 = {2 \rho ^2 + 2 \rho \over 2 \rho }-1 = \rho . \end{aligned}$$

We also have

$$\begin{aligned} \mathcal {C}'\left( \frac{\rho }{(1+\rho )^{2}} \right)= & {} {\mathcal {C}\left( \frac{\rho }{(1+\rho )^{2}} \right) ^2 \over 1 - 2 \frac{\rho }{(1+\rho )^{2}} \mathcal {C}\left( \frac{\rho }{(1+\rho )^{2}} \right) } = {(1+\rho )^2 \over 1 - 2 \frac{\rho }{(1+\rho )^{2}} (1+\rho )} = {(1+\rho )^3 \over 1 - \rho }. \end{aligned}$$

By putting everything together, we obtain

$$\begin{aligned} E(T_2(x_\rho ))= & {} \frac{(1-\rho ^2)}{\rho ^2(1+\rho )^{2}} \left( 1- \rho \ln \left( {1-\rho ^2\over \rho }\right) \right) \left[ \rho + {\rho \over (1+\rho )^2} {(1+\rho )^3 \over 1 - \rho } \right] \\= & {} \frac{(1-\rho ^2)}{\rho (1+\rho )^{2}} \left( 1- \rho \ln \left( {1-\rho ^2\over \rho }\right) \right) \left[ 1 + {1+\rho \over 1 - \rho } \right] \\= & {} \frac{2}{\rho (1+\rho )} \left( 1- \rho \ln \left( {1-\rho ^2\over \rho }\right) \right) . \end{aligned}$$