Macro quiz...well explained answers only

3.1 Exercise 4.1 Packets arrive to a communication node with a single output link according to a Poisson Process. Give the Kendall notation for the following cases: 1. the packet lengths are exponentially distributed, the buffer capacity at the node is infinite 2. the packet length is fixed, the buffer can store n packets 3. the packet length is L with probability pr and I with probability p and there is no buffer in the nodeIn a computer network a link has a transmission rate of C bit/s. Messages arrive to this link in a Poisson fashion with rate A messages per second. Assume that the messages have exponentially distributed length with a mean of 1/p bits and the messages are queued in a FCFS fashion if the link is busy. a) Determine the minimum required C for given A and p such that the average system time ( service time + waiting time) is less than a given time To.5.4 Exercise 5.7 Customers arrive to a single server system in groups of 1,2,3 anal .l customers. The number of customers per group is i.i.d. There are in total 4 places in the This is equivalent to the case where there is no queue and each customer is served in parallel with the others, so actually this system is equivalent to an Mfoano system! 1'? system. If a group of customers does not t into the system, none of the members of the group joins the queue. 13% of the customers arrive in a group of 1', 2t?% of the customers arrive in groups of El, 39% in a group of 3 and 49% in a group oft customers. The average number of arriving customers is 5'5 customers per hour, the interarrival time between groups is exponentially distributed. The service time is emponentially distributed with a mean of {15 minutes. I. Give the Kendall notation of the system and draw the state transition diagram. 2. Calculate the average number of customers in the queue and the mean waiting time per customer. 3. Calculate the probability that the system is full and the probability that a customer arriving in a group of h customers can not join the queue. 4. Calculate the probability that an arriving customer in general can not join the queue anal the probability that an arriving group of customers can not join the queue. 5. What is the average waiting time for a customer arriving in a group of 3 customers? 7 Exercise 6.3 We consider two types of call arrivals to a cell in a mobile telephone networlc: new calls that originate in a cell and calls that are handed over from neighboring cells. It is desirable to give preference to handover calls over new calls. For this reason, some of the channels in the cell are reserved for handover calls, while the rest of the channels are available to both types of calls. For the questions below, assume the following: Channels in the cell are held for two minutes on average, with exponential distribution. All calls arrive according to a Poisson process with rate A\": = 125 calls per hour for new calls, and Aha = 5|] calls per hour for handover calls. The cell has a capacity of Ill channels; each call occupies 1 channel. 1. Draw a state diagram of the channel occupancy in a cell when 2 channels are used exclusively for handover calls. 2. Calculate the blocking probability in the cell if no channels are reserved for handover calls. \"that is the average number of channels used .9 3. Find the minimum nurnber of channels reserved for handover calls so that their blocking probability is below 1 percent. What is the blocking probabil- ity for the new calls in this case? Consider a pure delay system where customers arrive according to a Poisson process with intensity A = 3. The service time is exponentially distributed with mean value 1/3. The queuing discipline is FCFS. There are two servers in the system, and one of them is always available. The other one starts service when the queue length would become two (so that it immediately becomes one). If there are no more customers in the queue, the server which becomes idle first is closed (and stays closed until the queue length becomes two again). Let us denote the state of the system with (ij) where i is the total number of customers in the system and j is the number of open servers. Give the Kendall notation of the system and draw the system diagram.In a kitchen dormitory corridor there are two hobs for cooking and 3 places on the sofa. There are 8 students living in the corridor, each of them goes on average every 1/a hours to the kitchen to cook (if he is not cooking already), the inter-arrival time is exponentially distributed. If on arrival the 2 hobs are occupied, the student looks for a place on the sofa. If the sofa is occupied as well, the student goes back to his room and tries again at a later time. Students spend an exponentially distributed amount of time cooking with mean 1/B. a = 0.5 hours, B = 1 hours. 1. Draw the state transition diagram 2. Calculate the mean waiting time of the students 3. Calculate the ratio of time the kitchen is completely full, e.g. a student arriving has to go back to his room. 4. Calculate the probability that a student finds the kitchen completely full 5. Calculate the probability that a student has to wait more than 2 hours (supposing that he can sit down in the kitchen)