Yuqian #3 - Queuing Theory 2 Flashcards
List 5 advantages of single lines
- Guarantees fairness
- No customer anxiety over queue choice
- Avoids “cutting in” issues
- More efficient for time minimisation
- Line switching/jockeying is avoided
List 4 advantages of a multiple line arrangement
- Service provided can be differentiated
- Labour specialisation is possible
- Customer has more flexibility
- Balking behaviour (when someone who otherwise would have entered a line decides not to) may be deterred
We can model a queue using Kendall’s Notation. How does this work? What does it look like?
List 5 assumptions that we make for a single-server model
- Infinite customer population
- Poisson distribution for customer arrival
- Exponential service distribution
- First come first served
- Unlimited length of line
A common type of queue model is the M/M/1 model. What does this represent?
A single-server model with Markovian (hence M) arrival process and service process - this means that the arrival process is modelled by a Poisson distribution while the service process is modelled by an exponential distribution.
Define the following notation related to queuing calculations:
Lq, L, Wq, W, and p
Lq - Average number of units in queue
L - Average number of units in system (waiting + being served)
Wq - Average waiting time in queue
W - Average time in system (waiting + being served)
p = Average utilisation of system
Lets say that units arrive at a repair shop according to a Poisson distribution at a rate of one per every 10 minutes, and the service time is exponentially distributed with a mean of 8 minutes. First come first served is being used.
(a) Find L, W, and Wq
(b) Repeat above, but arrival rate has increased by 10%
Formulae are provided below.
Little Muddy Creek has one launching ramp for small boats. On summer weekends, boats arrive for launching at a mean rate of 6 boats per hour. It takes an average of 6 minutes to launch a boat. Boats are launched FCFS. Find Lq, L, Wq, and W. Formulae are provided below.
How does waiting time vary with system utilisation for a Markovian queuing system?
What are the seven assumptions in an M/M/c queuing model? Hint: the first five are exactly the same as for an M/M/1 queuing model
- Infinite customer population
- Poisson distribution for customer arrival
- Exponential service distribution
- First come first serviced
- Unlimited length of line
- There are multiple servers, each serving customers
- If all c servers are busy, first customer in queue will be served as soon as a server becomes available
Another type of queuing system is the Jackson Network, where we analyse each server individually and then aggregate our results to find the system’s performance (except for time in system which requires Little’s Law). List 5 assumptions that must be satisfied for this type of queuing system.
- Independent and exponential inter-arrival time distribution
- Independent and exponential service time distribution
- Probabilistic routing
- Infinite queue capacity
- Utilisation is less than one for all servers
Little’s Law is used to calculate the time spent in the system for the Jackson Network. What is Little’s Law, and what is an important implication of this law?
Under steady-state conditions, the average number of items in the system is equal to the average rate of arrival multiplied by the average time spent in the system, or:
L= arrival rate * W
Lq = arrival rate * Wq
W = Wq + 1/(service rate)
Important implication: behaviour is independent of probability distributions, and requires no assumptions about schedule according to which customers arrive or are serviced
