Question

Consider the $\mathrm{H}_2 / \mathrm{M} / 1$ queueing system of Example 5.3.8. If $\tau$ has the distribution generated by Algorithm 3.2.2 show that $$ \pi_0=0.5-\rho+0.5 \sqrt{(1-2 \rho)^2+16 \rho q_1\left(1-q_1\right)(1-\rho)} . $$ 

   Consider the $\mathrm{H}_2 / \mathrm{M} / 1$ queueing system of Example 5.3.8. If $\tau$ has the distribution generated by Algorithm 3.2.2 show that
$$
\pi_0=0.5-\rho+0.5 \sqrt{(1-2 \rho)^2+16 \rho q_1\left(1-q_1\right)(1-\rho)} .
$$
Show moreā€¦
Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)
Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)
Arnold O. Allen 2nd Edition
Chapter 5, Problem 61 ā†“

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In the $\mathrm{H}_2 / \mathrm{M} / 1$ queueing system, we have a single server queue with a hyperexponential distribution for interarrival times and an exponential service time distribution. The parameter $\rho$ is the traffic intensity of the system, defined as  Show more…

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Consider the $\mathrm{H}_2 / \mathrm{M} / 1$ queueing system of Example 5.3.8. If $\tau$ has the distribution generated by Algorithm 3.2.2 show that $$ \pi_0=0.5-\rho+0.5 \sqrt{(1-2 \rho)^2+16 \rho q_1\left(1-q_1\right)(1-\rho)} . $$
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