5. Performance Evaluation and Models

Question 1

Distribution of Arrivals

Answer 1

• Generally assume arrival rate ~ Poisson(lambda)
• The number of customers coming into the system every period T is equal to lambda * T

Question 2

Distribution of Service Time

Answer 2

• Generally assume service time ~ exp(T_s)
• The average service time is mu = 1/T_s

Question 3

Notation - w

Answer 3

• The number of requests waiting in the queue

Question 4

Notation - q

Answer 4

• The total number of requests in the system

Question 5

Notation - T_s

Answer 5

• The average service time

Question 6

Notation - mu

Answer 6

• The average service rate
• 1 / T_s

Question 7

Notation - T_w

Answer 7

• The average waiting time in the queue

Question 8

Notation - T_q

Answer 8

• The average response time

Question 9

Notation - rho

Answer 9

• The utilization of the resource

Question 10

Steady State

Answer 10

• The rate at which requests are queued cannot exceed the rate at which the server can serve them
• lambda < mu
• rho < 1
• Arrival rate = throughput

Question 11

Little’s Law

Answer 11

• Under a steady state, the average number of items in a queuing system equals the average arrival rate multiplied by the average time an item spends in the queuing system.
• q = lambda * T_q
• w = lambda * T_w

Question 12

Relationship between Response Time, Waiting Time, and Service Time

Answer 12

• T_q = T_w + T_s

Question 13

Relationship between total items in the system, items waiting, and utilization

Answer 13

• q = w + rho

Question 14

Utilization (Steady-State)

Answer 14

• rho = lambda * T_s (lambda must be less than 1/T_s)