Solutions to Chapter 4 - Communication Networks

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Communication Networks (2 nd Edition) Chapter 4 Solutions 7 9 0.362 7 90 0.923 8 9 0.289 . . . 9 9 0.224 . . . 10 9 0.168 100 90 0.027 As shown, a lower probability of blocking is achieved by scaling up the system. 59. Calls arrive to a pool of 50 modems according to a Poisson process. Calls have an average duration of 25 minutes. Solutions follow questions: a. What is the probability an arriving calls finds all modems busy if the arrival rate is two calls per minute For this system, the utilization a = 2/(1/25) = 50 and c = 50. The probability of blocking is: P b = c a c! c a j j j= 0 ! = 0.105 b. What is the maximum arrival rate that can be handled if the maximum acceptable blocking probability if 1% 10% If the maximum acceptable blocking probability is 10% and 1%, the maximum handled arrival rates, given a service rate of 1/25 calls per minute are: P b Load 1% 1.516 calls/minute 10% 1.982 calls/minute 60. Consider dynamic nonhierarchical routing (DNHR). Solutions follow questions: a. Explain how DNHR can be used to exploit the time differences between different time zones in a continent. In the morning in the east, the business day begins and long-distance calls are made between cities on the East Coast. The west-coast business day has not yet begun, so the links from east to west are relatively under-utilized. Eastern traffic can be re-routed through the western free lines to provide additional capacity for the traffic between eastern cities. The opposite can occur in the evening after the business day in the east ends and the under-utilized eastern trunks are used for western traffic. b. Explain how DNHR can be used to exploit different business and residential activity patterns during the day. For local calls during business hours, traffic may be re-routed through relatively non-busy lines in residential area to free up resources in the business areas. The opposite can be done in the evening when most people are in the residential areas and the telephone traffic in the business area is low volume. Leon-Garcia/Widjaja 28
Communication Networks (2 nd Edition) Chapter 4 Solutions 61. Suppose that setting up a call requires reserving N switch and link segments. a. Suppose that each segment is available with probability p. What is the probability that a call request can be completed For a call to be completed, every segment in the circuit must be free. Thus, assuming that each switch and link has a probability p of being available, and assuming that this probability is independent of all others, P[every circuit free] = P[1 st free]P[2 nd free]…P[N th free] = p N b. In allocating switch and transmission resources, explain why it makes sense to give priority to call requests that are almost completed rather than to locally originating call requests. For calls that are almost completed, resources are already locked up, so if the call is denied, there is a large overhead penalty. Thus they should be given priority. To illustrate, suppose locally originating calls are given a higher priority. One can then envision a situation where very few completed connections exist, but all resources are used by partially completed calls. 62. Consider a cellular telephone system with the following parameters: B is the total bandwidth available for the system for communications in both directions; b is the bandwidth required by each channel, including guard bands; R is the re-use factor; and a is the fraction of channels used for set up. Solutions follow questions: a. Find an expression for the number of channels available in each cell. Number of channels available in each cell = B ( 1− a) b R channels/cell b. Evaluate the number of channels in each cell for the AMPS system. In AMPS, there are 416 total channels, 21 of which are used for call setup. The reuse factor is 7. Thus the number of channels per cell is: (416 –21)/7 = 56 channels per cell 63. Consider the AMPS system in problem 62. Solutions follow questions: a. How many Erlangs of traffic can be supported by the channels in a cell with a 1% blocking probability 5% The probability of blocking is given by the equation: P b = c a c! c a j j j= 0 ! where in the AMPS system c = 56 and a is the number of Erlangs of traffic in a cell. If the blocking probability is 10% and 1%, the maximum number of Erlangs that can be handled is: Leon-Garcia/Widjaja 29
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Solutions to Chapter 4 - Communication Networks

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