Telecommunications System Modelling

You are responsible for the planning of roads in a small suburb shown in Figure below. There are 4 intersections, labelled 1 to 4. The outside of this suburb has been represented by areas 0 and 5. Each street segment is named based on the two intersections that it connects to in the direction of traffic flow. For example, s₁₂ is the road segment connecting intersection 1 to intersection 2. The opposite direction would be called s₂₁.

Similarly, s₀₁ connects the left-hand outside area 0 to intersection 1, and s₃₅ connects the intersection 3 to the outside area 5 at the bottom. Some example names for streets are shown on the figure.

When cars arrive at an intersection, a percentage q_L of them turn left, q_R turn right, and q_S go straight. Also, a percentage q_E will leave the street network (going home or to their office). These percentages are the same for all intersections. At the edge of the suburb, the turning cars will re-enter the suburb at the next available street and the ones going straight are recycled back from the other end as shown in the Figure. This is meant to model arrivals from other neighbouring suburbs to this area.

The cars arrive to the system (from houses or offices) at a rate of ?? in each street segment (same rate for all segments) according to a Poisson process.

To model the queue at the intersection, you can use an M/M/k queue, where k is the number of lanes in the street. The cars depart the intersection at the rate of μ cars per second for each lane. In other words, μ is the service rate of each lane (each lane is a server in the terminology of queueing theory). If k > 1 in any intersection, it means that cars have more than one lane so this becomes a multi-server queue. Of course, making streets wider to have more than one lane is very costly, so the aim is to identify the minimum number of lanes that are sufficient to give a reasonable congestion delay for the commuters. Your tasks are as follows:

1- Model this street network as a Jackson queuing network. Identify all relevant parameters of the system.

2- Write down the expressions for the total average delay (W) in the system, as well as other relevant indicators such as the size of queues at intersections. Clearly state any assumptions that you have to make.

3- Discuss how W is related to k. It is expected that as k increases, then the congestion delay is reduced. Is this correct?

4- Solve this problem and obtain W (if a steady state solution exists) for the following numerical example: γ = 2, μ = 8, q_L = q_R = 0.2, q_S = 0.4, q_E = 0.2, k = 2. Use Matlab for this purpose and attach your code. Discuss the solution, if it exists or if you needed to make more assumptions in derivation.

5- What range of values of μ will result in a steady state solution for k = 1.

You need to submit a hardcopy report for the above five tasks. Don't write an essay, make it brief and clear. I expect the report to be a maximum of 3 pages (but it is ok if you go a bit above this). Also attach the printout of your code and evidence that you have run the code and obtained the results in part 4.