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I am stuck on trying to figure out how to get started? Can someone maybe push me in the right direction?

A bank that is planning to install an ATM must choose between buying one Zippytel machine or two Klunkytel machines. Although one Zippy costs twice as much to purchase, install, and operate as one klunky, the Zippy works twice as fast. Since the total cost to the bank is the same regardless of the decision, the mangers would like to install the system that will provide the best service. From available data, it appears that during a certain rush period, customers arrive one at a time according to a Poisson process, with rate 1 per minute. Zippy service times are exponentially distributed with mean of 40 seconds. Klunky service times are exponentially distributed with mean 80 seconds.

I need to use continuous time Markov Chains to Model the system. I am just lost on where to start.

daultongray8
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1 Answers1

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Ok, you have two situations: 1) One Zippy ATM with the customers' mean interarrival time equalling 1 min and mean service time equalling 0.4 min. 2) Two Klunky ATMs with customers' mean interarrival time equalling 1 min and mean service time equalling 0.8 min for each.

The first situation is a M\M\1 queue (markovian queue with 1 server) with $\lambda=1$, $\mu=\frac{1}{0.4}=2.5$ and traffic intensity $p=\frac{\lambda}{\mu}=0.4$. You can model it as a birth-death process with $\lambda_n=\lambda$ and $\mu_n=\mu$. Traffic intensity $p<1$ so it has steady state distribution (probability of occuring in state n, meaning n number of customers as $t \to \infty$): $u_n=\frac{\pi_n}{\sum_{n=0}^{\infty}\pi_n}=\frac{\frac{\lambda_0\lambda_1...\lambda_{n-1}}{\mu_1\mu_1...\mu_{n}}}{\sum_{n=0}^{\infty}\frac{\lambda_0\lambda_1...\lambda_{n-1}}{\mu_1\mu_1...\mu_{n}}}=\frac{(\frac{\lambda}{\mu})^n}{\sum_{n=0}^{\infty}(\frac{\lambda}{\mu})^n}=\frac{p^n}{\sum_{n=0}^{\infty}p^n}=(1-p)p^n$.

The second situation is a M\M\s (markovian queue with s servers) where $s=2$, $\lambda=1$, $\mu=\frac{1}{0.8}=1.25$ and $p=0.8<1$. It can alse be modelled as a birth-death process with $\lambda_n=\lambda$ and $\mu_0=0$, $\mu_n=n\mu$ for $n \in [1, s]$ and $\mu_n=s\mu$ for $n>s$.

From these steady state distributions you can calculate mean waiting time of a customer (or better expected sojourn time of a customer = waiting time plus service time) for both queues and compare them.

Koncopd
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