# D/D/1 Queuing

1. At the end of a sporting event, vehicles begin leaving a parking lot at λ(t) = 12 − 0.25t and vehicles are processed at μ(t) = 2.5 + 0.5t [t is in minutes and λ(t) and μ(t) are in vehicles per minute]. Assuming D/D/1 queuing, determine the total vehicle delay, longest queue, and the wait time of the 50th vehicle to arrive assuming first-in-first-out and D/D/1 queuing.

2. Vehicles arrive at a single park-entrance booth where a brochure is distributed. At 8 A.M., there are 20 vehicles in the queue and vehicles continue to arrive at the deterministic rate of λ(t) = 4.2 − 0.1t, where λ(t) is in vehicles per minute and t is in minutes after 8:00 A.M. From 8 A.M. until 8:10 A.M., vehicles are served at a constant deterministic rate of three per minute. Starting at 8:10 A.M., another brochure-distributing person is added and the brochure-service rate increases to six per minute (still

at a single booth). Assuming D/D/1 queuing, determine the longest queue, the total delay from 8 A.M. until the queue dissipates; and the wait time of the 40th vehicle to arrive.

3. Vehicles arrive at a single toll booth beginning at 7:00 A.M. at a rate of 8 veh/min. Service also starts at 7:00 A.M. at a rate of μ(t) = 6 + 0.2t where μ(t) is in vehicles per minute and t is in minutes after 7:00 A.M. Assuming D/D/1 queuing, determine when the queue will clear, the total delay, and the maximum queue length in vehicles.

4. Vehicles begin arriving at a single toll-road booth at 8:00 A.M. at a time-dependent deterministic rate of λ(t) = 2 + 0.1t [with λ(t) in veh/min and t in minutes]. At 8:07 A.M., the toll booth opens and vehicles are serviced at a constant deterministic rate of 6 veh/min. Assuming D/D/1 queuing, what is the average delay per vehicle from 8:00 A.M. until the initial queue clears and what is the delay of the 20th vehicle to arrive?

5. Vehicles begin to arrive at a toll booth at 8:50 A.M. with an arrival rate of λ(t) = 4.1 + 0.01t [with t in minutes and λ(t) in vehicles per minute]. The toll booth opens at 9:00 A.M. and processes vehicles at a rate of 12 per minute throughout the day. Assuming D/D/1 queuing, when will the queue dissipate and what will be the total vehicle delay?

6. Vehicles begin to arrive at a toll booth at 7:50 A.M. with an arrival rate of λ(t) = 5.2 − 0.01t [with t in minutes after 7:50 A.M. and λ in vehicles per minute]. The toll booth opens at 8:00 A.M. and serves vehicles at a rate of μ(t) = 3.3 + 2.4t (with t in minutes after 8:00 A.M. and μ in vehicles per minute). Once the service rate reaches 10 veh/min, it stays at that level for the rest of the day. If queuing is D/D/1, when will the queue that formed at 7:50 A.M. be cleared?

7. Vehicles arrive at a freeway on-ramp meter at a constant rate of six per minute starting at 6:00 A.M. Service begins at 6:00 A.M. such that μ(t) = 2 + 0.5t, where μ(t) is in veh/min and t is in minutes after 6:00 A.M. What is the total delay and the maximum queue length (in vehicles)?

8 Vehicles arrive at a toll booth according to the function λ(t) = 5.2 − 0.20t, where λ(t) is in vehicles per minute and t is in minutes. The toll booth operator processes one vehicle every 20 seconds. Determine total delay, maximum queue length, and the time that the 20th vehicle to arrive waits from its arrival to its departure.

9. There are 10 vehicles in a queue when an attendant opens a toll booth. Vehicles arrive at the booth at a rate of four per minute. The attendant opens the booth and improves the service rate over time following the function μ(t) = 1.1 + 0.30t, where μ(t) is in vehicles per minute and t is in minutes. When will the queue clear, what is the total delay, and what is the maximum queue length?

10. Vehicles begin to arrive at a parking lot at 6:00 A.M. with an arrival rate function (in vehicles per minute) of λ(t) = 1.2 + 0.3t, where t is in minutes. At 6:10 A.M., the parking lot opens and processes vehicles at a rate of 12 per minute. What is the total delay and the maximum queue length?

11.At a parking lot, vehicles arrive according to a Poisson process and are processed (parking fee collected) at a uniform deterministic rate at a single station. The mean arrival rate is 4.2 veh/min and the processing rate is 5 veh/min. Determine the average length of queue, the average time spent in the system, and the average waiting time in the queue.

12.Consider the parking lot and conditions described in Problem 5.27. If the rate at which vehicles are processed became exponentially distributed (instead of deterministic) with a mean processing rate of 5 veh/min, what would be the average length of queue, the average time spent in the system, and the average waiting time in the queue?

13Trucks begin to arrive at a truck weigh station (with a single scale) at 6:00 A.M. at a deterministic but time-varying rate of λ(t) = 4.3 − 0.22t [λ(t) is in veh/min and t is in minutes]. The departure rate is a constant 2 veh/min (time to weigh a truck is 30 seconds). When will the queue that forms be cleared, what will be the total delay, and what will be the maximum queue length?

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