1. Introduction Notation Little’s Law aka Little’s Result
Queueing theory
Introduction
Notation
Little’s Law aka Little’s Result

2. Notation/Definitions in Queuing Theory
 - means the number of arrivals per second
 - service rate of a device
T - mean service time for each arrival
 - Utilization, percentage of time a server is in use
Q - number of customers in the system waiting or being served
N - number of servers
Response time - time from when a process is entered the queue, until it completes

3. Service Rate = 1/Service Time
A cashier can serve 1 customer every 3 minutes
- The service time = 3 minutes
The service rate is
1 customer every 3 minutes. or
1/3 a customer per minute
Ẋ = 3 minutes and µ=1/3 per minute
Note a rate is an Amount/Time and a service time is an (amount of time)/service

4. Little’s Law (a.k.a. Little’s Result)
 = arrival rate.
𝑊 𝑞 = mean time a process spends in a queue.
𝑁 𝑞 = number of processes in the queue.
𝑊 𝑠 = mean time a process spends in the system
𝑁 𝑠 = number of processes in the system.
𝑁 𝑠 =  𝑊 𝑠 and 𝑁 𝑞 =  𝑊 𝑞
Q1) 10 people walk into a supermarket every minute. On average, they stay for 30 minutes. How many customers are in the supermarket at any random time?
Answer) N = (10/minute) * ( 30 minutes) = 300
Q2) are there always exactly 300 customer in the supermarket?

5. Interarrival distributions - Deterministic
Suppose a bus company has buses arrive every hour on the hour at a particular bus stop. The buses are never late and never early…they are always on time.
The arrival rate λ = 1/hour
The distribution is Deterministic.
Interesting questions
If the last bus arrived 20 minutes ago, when is the next bus coming?
If you arrived at the bus stop at an arbitrary time, how long on average would it be until the next bus arrives?

6. Interarrival distributions - Markovian
Suppose a roulette wheel has 38 numbers (0, 00, 1..36) on it and your favorite number is 15.
The arrival rate λ = 1/38 spins
The distribution is Poisson (type of Markovian).
Interesting questions
If the last #15 arrived 20 spins ago, when is the next #15 coming?
If you arrived at the roulette table at an arbitrary time, how many spins would it be until the next 15 arrives?

7. Expected number of spins
There must be at least 1 spin and then there is a probability that a 15 doesn’t come up and you have to start the whole game over.
E = 1 + (37/38) E
(1/38)E = 1
E= 38

8. D/D/1 queue with 𝜆< 𝜇 Example 𝜆=2/𝑠𝑒𝑐 deterministically
𝜇=3/sec deterministically
𝜌= 𝜆 𝜇 = 2 3
Arrivals
Service completions

9. M/M/1 queue with 𝜆< 𝜇 Example 𝜆=2/𝑠𝑒𝑐 Markivian 𝜇=3/sec Markovian
𝜌= 𝜆 𝜇 = 2 3
Arrivals
Service completions

10. Kendall notation A/B/c/K/m/z A – interarrival distribution
B – service time distribution
c – number of servers
K – the capacity of the queue
m – number of customers in the system
z – queuing discipline

11. Examples of Kendall notation
A/B/c is used when
There is no limit on the length of the queue = default infinite queue
The source is infinite – open system
The queue discipline is FCFS (default)
A and B may be any of the following:
GI for general independent interarrival time
G for general service time
E for Erlang-k interarrival or service time distribution
M for exponential interarrival or service time distribution
D for deterministic interarrival or service time distribution
H or hyperexponential (with k stages) interarrival or service time distribution.

12. M/D/3/4/12/FCFS

13. Definitions in Queuing Networks
Response time – time from when a process is entered the network of queue, until it leaves the network.
Throughput – Work / Time
Bottlenecks – a resource that has the highest server utilization
Saturated System – a system where the bottleneck device has reached 100% utilization.

14. M/M/1 Queue Example
Consider the arrival rate  = 1/second and the service rate  = 1/second, what would the:
Utilization 𝜌 of the server be?
What would be the mean time in the system Ts ?
What would be the mean time in queue Tq ?
Institution might lead you to believe that since arrivals are 1 per second and the time to service them is 1 per second, therefore every customer arrives exactly 1 second after the previous customer goes straight to the server (no waiting in the queue) each customer will be serviced in exactly 1 so therefore, the utilization is 100% and the time in the queue is 0 an the time in the system is 1 second.

15. GI/G/3 queue What is the average utilization of each server?
𝜆= 𝜆1+ 𝜆2=2/ sec + 1/ sec =3/𝑠𝑒𝑐
𝜇=4 𝑝𝑒𝑟 𝑠𝑒𝑟𝑣𝑒𝑟
𝜌= 𝜆 𝑐𝜇 = 3 (3)(4) = 25%
𝜇=4/𝑠𝑒𝑐
𝜇=4 sec
𝜇=4/sec
𝜆 =2/𝑠𝑒𝑐
𝜆 =1/𝑠𝑒𝑐