Anthony Khoueiry: At a defined position of a paid highway, all lanes are joined into one lane where a camera is installed to photoshoot the plates of the vehicles in order to automatically withdraw the tax of the highway from their accounts. It has been estimated that an hourly capacity of 1500 passengers can be served with the existing camera. Due to future expansion plans if the area, countermeasures are taken in a way that one third of the entering traffic is prohibited for 2 hours, to ensure the infrastructure works. After two hours, all the capacity (1500 veh/hr) is allowed to enter to this segment.
A count of recent passenger reveals an arrival function as shown below:
λ = 1500 for 0λ = 500 for t>1
λ is the arrival function and t the time in hours.
The queuing model is assumed to be D/D/1, so that the arrival and the departure functions are linear. 

Hint: the arrival are noted as reaching the camera position, and the departure are noted as beign photo shooted by the camera. The photo duration is instant and its duration is negligible.
[20/03, 11:22 am] Anthony Khoueiry: a) Write the function of departure μ, the equations of the total arrivals and the equations of the total departures, as function of time.
b) Represent graphically the arrival and the departure function on this segment.
c) By calculations, show that there is no possibility that the que will dissipate at a time t less than 1 hour. (2 pts)
By calculations also, find the time (t in hours) at which the queue will dissipate. (3 pts)
d) How many cars will have arrived and departed at this time. (2 pts)
e) Define and find graphically the maximum queue length. (2 pts)
f) Which vehicle will have the longest delay and find the duration of this delay. (4 pts)
g) Define and find graphically the total vehicle delay. (2 pts)
h) Find the average queue length. (2 pts)
i) Find the average delay time. (2 pts)
j) Find by analytical calculation the delay of a passenger arriving 30 minutes after the start application of the countermeasures knowing that as noted above, these countermeasures take 2 hours. Check your result graphically

Answer rating: 100% (QA)

Quiz a The departure function is given by t 1500 for t 1 t 500 for t1 The arrival fu... View the full answer