3.4 Give correct answers

0.5m 0.5mr 1. A M/-/1/2 queue has a service time distribution given by The s + m (s+ m)2 average arrival rate is 1. Note that the queue is limited to a maximum state of 2. Use the method of stages to solve this queue and obtain the following - (a) State Transition Diagram (with a proper definition of system states) 13] (b) Obtain the state probability distribution [6] (c) What will be the average departure rate from this queue? 13] 2. Consider a MIXI/-/1/3 queue where the batch arrival rate is I and the generating function of the batch sizes is given by (0.25 +0.25z +0.5z ). Note that the queue is limited to a maximum state of 3 and follows the WBAS strategy (a) If the service time distribution is 0.5m 0.5m draw the state transition diagram s+ m (s+ m)- of the queue with an appropriately defined system state. 13] (b) For a service time distribution of m do the following - s+ m i. Draw the state transition diagram. ii. Find the state probabilities of the queue. [5] iii. What is the probability that a batch is refused entry into the queue? (2]0.5m 0.5mr 1. A M/-/1/2 queue has a service time distribution given by The s+m (s+ m)- average arrival rate is 1. Note that the queue is limited to a maximum state of 2. Use the method of stages to solve this queue and obtain the following - (a) State Transition Diagram (with a proper definition of system states) 13] (b) Obtain the state probability distribution [6) (c) What will be the average departure rate from this queue? [3] 2. Consider a MIXI/-/1/3 queue where the batch arrival rate is I and the generating function of the batch sizes is given by (0.25 +0.25z +0.5z'). Note that the queue is limited to a maximum state of 3 and follows the WBAS strategy (a) If the service time distribution is 0.5m 0.5m draw the state transition diagram s+m (s+ m)- of the queue with an appropriately defined system state. 13] (b) For a service time distribution of -, do the following - s+ m i. Draw the state transition diagram. ii. Find the state probabilities of the queue. iii. What is the probability that a batch is refused entry into the queue? (2]1. Consider an M/G/1 queue at equilibrium, where the server goes on a single vacation of random length whenever the system becomes empty. Let f, () be the pdf of the length of the vacation period with L.S.T. F, (s) and with V and V? as its first and second moments respectively. Use the Imbedded Markov Chain approach to find- (a) The probability po that the system is empty (3 +3) (b) The Generating Function P(z) of the number in the system (6) [Note: 3 marks in (a) above are for writing the correct equation(s) describing the Imbedded Markov Chain] 2. Consider an M/G/1 system with exceptional first service. We are given that the first and second moments of a normal service time are X and X? , respectively. However, the first and second moments of the first service duration in a busy period are & and X? . respectively. Use the Residual Life approach to find the mean waiting time in queue W and the mean time spent in system W for a job arriving to the system under equilibrium conditions. (9+4)