a) i) Using appropriate diagrams, explain the difference between probability mass function and probability density function.
ii) Explain the definition of subsystem reliability, and the use of the exponential distribution in this context. What are the advantages of the exponential distribution here? Considering an electronic circuit, suggest an element that might be a “subsystem”.
iii) A Binomial distribution is used to model the results of 1000 experiments. If the standard deviation of this Binomial distribution is 0.1, what is the success probability? Suggest an example of where the Binomial distribution could be used in evaluating digital communications. [9 marks]
- b) A queue model is used to model the number of jobs queueing for a microprocessor. We assume jobs arrive as a Poisson process at a rate of 1 job per millisecond, and that jobs take an exponentially distributed amount of time for service. At what rate will the microprocessor have to “service” these jobs in order to ensure that:
i) the mean number of jobs in the queue is < 2?
ii) the probability that there are >10 jobs in the queue is < 0.001? [8 marks]
c) A computer C0 is acting as a packet router. It is connected to 100 other computers (C1→C100). All of C1→C100 communicate to the internet via computer C0.
During a busy hour, each of the computers C1→C100 generates an average of 10 packets per second, randomly, to upload to the internet via C0. Each of these packets passes through the packet queue in a packet buffer in C0.
We assume every packet is fixed at 1000bytes in length.
Sketch a diagram of this networking scenario and propose a simple queue model for the packet buffer in C0.
Given this scenario we decide the mean wait in the packet buffer at C0 must not exceed 10ms. What minimum bandwidth link is required to connect C0 to the wider internet to ensure that the mean wait in the packet buffer does not exceed 10ms?
(Use the simple queue model you proposed.)
If we decide we wish to reduce the mean waiting time by an order of magnitude,recalculate the required packet service rate at C0. Does this require a significant increase in packet service rate (bandwidth) at C0? [8 marks]
a) i) Explain how the Erlang B and Erlang C queues are used to model call centres.
Which one models which scenario? State the assumptions concerning call arrival process and service process.
ii) A call centre has 100 people working on the phones at busy periods. Assume there is in effect infinite waiting space for people to queue while they wait. The applied traffic is 90 Erlangs at busy periods. What is the probability that you have to wait >10 seconds in the queue, given that you do have to wait? Assume that the mean call service time (holding time) is 10 minutes.
iii) Draw a fully labelled diagram of a Discrete Time Discrete Space (DTDS) Markov Chain (MC) model for a communications channel with these characteristics:
Good state SNR = 50dB, bad state SNR = 10dB At the end of each second the MC transitions to the good state w.p. 0.01 (if it was in the bad state). If it is in the good state it stays there w.p. 0.999 [9 marks]
b) A system consists of hardware, replicated 3 times (i.e. there are a total of 4 versions of the hardware), and software, which we assume is common to all
hardware. Draw a diagram of the reliability model of such a system.
Evaluate the system reliability after 1 year given the fail rates are as follows:
Software: 1 failure per month Hardware subsystems: fail rate is 6 per year per subsystem. [8 marks]
c) You are working on designing a communications channel. You guess from the electronics of the system that the Bit Error Rate (BER) should be about 10 errors in 10,000 bits (BER=0.001). Explain how you would systematically estimate the BER and find bounds (confidence intervals) on its value.