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\$$\\lambda=\\var{lambda}\$$  and  \$$\\mu=\\var{mu}\$$

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P(operator is busy) = 1 - P(no calls in the system)

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= 1 - \$$P_0\$$

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= 1 - \$$\\frac{\\lambda}{\\mu}\$$

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= 1 - \$$\\var{p_0}\$$

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Percentage probability = \$$\\simplify{1-{p_0}}*100\$$%

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= \$$\\simplify{(1-{p_0})*100}\$$%

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P(No calls in queue) = P( 0 calls in system OR 1 call in the system)

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= \$$P_0+P_1\$$

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= \$$\\var{p_0}+\\var{p_1}\$$

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P(at least n calls in queue) = P(at least n+1 calls in the system)

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= 1 - P(less than n+1 calls in the system)

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= 1 - {\$$P_0+P_1+....P_n\$$}

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The mean time a customer will wait for service = \$$W_Q=\\frac{\\lambda}{\\mu(\\mu-\\lambda)}\$$

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", "statement": "

A hotel telephone exchange employs one operator to connect incoming and outgoing calls.

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Calls arrive at a mean rate of \$$\\var{lambda}\$$ calls per minute. The mean service rate is \$$\\var{mu}\$$ calls per minute.

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Calculate the percentage probability that the operator is busy. [[0]]

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Calculate the probability that there are no calls queueing for service. [[1]]

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Calculate the probability that there are at least \$$\\var{n}\$$ calls in the queue. [[2]]

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Calculate the mean time (minutes) a customer will have to wait for service. [[3]]