Queueing

Question 1

population arrival

Answer 1

infinite generation following a poisson distribution: random but with a fixed mean rate. interarrival times follow an exponential time

Question 2

queue

Answer 2

where the population wait before being served
discipline, how someone from the queue is selected, random, first (usual way), …
queue length is number waiting

Question 3

servers

Answer 3

serve the population, can be in parallel (then completely served by it) and can in sequence (going from one server to another). here usually its 1 single sequence system
the time stayed at a server is the service time, usually exponential distribution. here independent and identically distributed

Question 4

state of system

Answer 4

n population in system

Question 5

steady state

Answer 5

at first system affected by initial state: transient condition
then with sufficient time it reaches steady state

Question 6

rate diagram

Answer 6

the states are the number of people in the queue. the more people can be in the system & queue the longer you make it
arrow in are the numbers entering, arrow out are the numbers leaving
the servers should be at the beginning, 1 node 1 server and how many can handle, 2nd node 2nd server and how many can handle together with 2 and so on, the last nodes are the waiting nodes (with the same arrows as the last server)

Question 7

computing stationary distribution from the rates

Answer 7

rate in = rate out for each state
=> rate * pi where arrow comes from
sum rates = 1

there will always be one redundant equation that you can remove (can cancel terms from each other by comparing equations)

try to solve equations by reducing terms and having only 1 remaining unknown pi for all (the same one for each equation). eg: pi_1= 3/2 pi_0 & pi_2 = 1/2 pi_0

Put all fraction over the same denominator X. The remaining pi will be X/X, eg pi_0= 2/2 . Sum numerator for all pi to find new denominator which make sum of fraction = 1, eg 3+1+2 = 6 => pi_0=2/6,pi_1=3/6,pi_2=1/6.
Now put all fractions found in pi=(pi_1,pi_2,…), eg: (2/6,3/,6,1/6)

Question 8

getting probas(compute time in each state then compute stationary with them) from rates

Answer 8

transform the rates such that the sum of transition from each node = 1
time spent in a node: unit of time/ how many leave(go to next and previous)
take the probas and multiply by expected time at node

Question 9

how many expected people in the system

Answer 9

sum_i pi_i * i

Question 10

how long will it take to reach y from x

Answer 10

compute the probas (transitions from x to i. Do fractions, customers leaving to i / total customers leaving from x)
compute how much time spent at each node (full queue time / people leaving)
do Ex equations ( proba of going to state i from x * E_i), instead of 1 + E do time spent at node + E
solve for Ex and have Ey = 0