Probability of highest and lowest of a vehicle arrival. Concept introduction: The M/M/1 queue assumes poison arrivals, exponential service times, and a single server serving customers in a FCFS fashion. Poisson arrivals are a reasonably good assumption for unscheduled systems. Further if there is a mix of many different types of jobs the exponential distribution can be realistic for service times. Otherwise it tends to be too variable of a distribution.

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Answer 1

Question

Chapter 5, Problem 3.3Q

Summary Introduction

Interpretation: Probability of highest and lowest of a vehicle arrival.

Concept introduction:The M/M/1 queue assumes poison arrivals, exponential service times, and a single server serving customers in a FCFS fashion. Poisson arrivals are a reasonably good assumption for unscheduled systems. Further if there is a mix of many different types of jobs the exponential distribution can be realistic for service times. Otherwise it tends to be too variable of a distribution.

Expert Solution & Answer

Answer to Problem 3.3Q

Highest in overnight hours and lowest in commuters’ hours.

Explanation of Solution

Given information:

average cars per hour(λ)=100per hourservice rate(μ) = 240 per hourno.of vehicles in the queue = Lq≤4 traffic rate(ρ)=(λμ) =100240 =0.42no: of vehicles in queue(Lq) =ρ21−ρ =0.30 ρ0=1−ρ =1−0.42 =0.58..........................2

(references from 97136-5-3.1Q)

λ=75 per hourμ=240 per hourno.of vehicles in the queue = Lq≤4ρ=( 75 240)=0.3125Lq=ρ2 1−ρ =0.142 P 0 =1−ρ =1−0.3125 =0.6875

0.103 [References from 97136-5-3.1Q]

0.58 [References from 97136-5-3.2Q]

0.6875 [References from 97136-5-3.3Q]

Hence, the probability that a vehicle finds a lane idle and gets instant service is highest for the overnight hours and lowest for the commuter hours.

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Answer 2

Question

Chapter 5, Problem 3.3Q

Summary Introduction

Interpretation: Probability of highest and lowest of a vehicle arrival.

Concept introduction:The M/M/1 queue assumes poison arrivals, exponential service times, and a single server serving customers in a FCFS fashion. Poisson arrivals are a reasonably good assumption for unscheduled systems. Further if there is a mix of many different types of jobs the exponential distribution can be realistic for service times. Otherwise it tends to be too variable of a distribution.

Expert Solution & Answer

Answer to Problem 3.3Q

Highest in overnight hours and lowest in commuters’ hours.

Explanation of Solution

Given information:

average cars per hour(λ)=100per hourservice rate(μ) = 240 per hourno.of vehicles in the queue = Lq≤4 traffic rate(ρ)=(λμ) =100240 =0.42no: of vehicles in queue(Lq) =ρ21−ρ =0.30 ρ0=1−ρ =1−0.42 =0.58..........................2

(references from 97136-5-3.1Q)

λ=75 per hourμ=240 per hourno.of vehicles in the queue = Lq≤4ρ=( 75 240)=0.3125Lq=ρ2 1−ρ =0.142 P 0 =1−ρ =1−0.3125 =0.6875

0.103 [References from 97136-5-3.1Q]

0.58 [References from 97136-5-3.2Q]

0.6875 [References from 97136-5-3.3Q]

Hence, the probability that a vehicle finds a lane idle and gets instant service is highest for the overnight hours and lowest for the commuter hours.

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 3

Question

Chapter 5, Problem 3.3Q

Summary Introduction

Interpretation: Probability of highest and lowest of a vehicle arrival.

Concept introduction:The M/M/1 queue assumes poison arrivals, exponential service times, and a single server serving customers in a FCFS fashion. Poisson arrivals are a reasonably good assumption for unscheduled systems. Further if there is a mix of many different types of jobs the exponential distribution can be realistic for service times. Otherwise it tends to be too variable of a distribution.

Expert Solution & Answer

Answer to Problem 3.3Q

Highest in overnight hours and lowest in commuters’ hours.

Explanation of Solution

Given information:

average cars per hour(λ)=100per hourservice rate(μ) = 240 per hourno.of vehicles in the queue = Lq≤4 traffic rate(ρ)=(λμ) =100240 =0.42no: of vehicles in queue(Lq) =ρ21−ρ =0.30 ρ0=1−ρ =1−0.42 =0.58..........................2

(references from 97136-5-3.1Q)

λ=75 per hourμ=240 per hourno.of vehicles in the queue = Lq≤4ρ=( 75 240)=0.3125Lq=ρ2 1−ρ =0.142 P 0 =1−ρ =1−0.3125 =0.6875

0.103 [References from 97136-5-3.1Q]

0.58 [References from 97136-5-3.2Q]

0.6875 [References from 97136-5-3.3Q]

Hence, the probability that a vehicle finds a lane idle and gets instant service is highest for the overnight hours and lowest for the commuter hours.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Question

Chapter 5, Problem 3.3Q

Summary Introduction

Interpretation: Probability of highest and lowest of a vehicle arrival.

Concept introduction:The M/M/1 queue assumes poison arrivals, exponential service times, and a single server serving customers in a FCFS fashion. Poisson arrivals are a reasonably good assumption for unscheduled systems. Further if there is a mix of many different types of jobs the exponential distribution can be realistic for service times. Otherwise it tends to be too variable of a distribution.

Expert Solution & Answer

Answer to Problem 3.3Q

Highest in overnight hours and lowest in commuters’ hours.

Explanation of Solution

Given information:

average cars per hour(λ)=100per hourservice rate(μ) = 240 per hourno.of vehicles in the queue = Lq≤4 traffic rate(ρ)=(λμ) =100240 =0.42no: of vehicles in queue(Lq) =ρ21−ρ =0.30 ρ0=1−ρ =1−0.42 =0.58..........................2

(references from 97136-5-3.1Q)

λ=75 per hourμ=240 per hourno.of vehicles in the queue = Lq≤4ρ=( 75 240)=0.3125Lq=ρ2 1−ρ =0.142 P 0 =1−ρ =1−0.3125 =0.6875

0.103 [References from 97136-5-3.1Q]

0.58 [References from 97136-5-3.2Q]

0.6875 [References from 97136-5-3.3Q]

Hence, the probability that a vehicle finds a lane idle and gets instant service is highest for the overnight hours and lowest for the commuter hours.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Chapter 5, Problem 3.3Q

Summary Introduction

Interpretation: Probability of highest and lowest of a vehicle arrival.

Concept introduction:The M/M/1 queue assumes poison arrivals, exponential service times, and a single server serving customers in a FCFS fashion. Poisson arrivals are a reasonably good assumption for unscheduled systems. Further if there is a mix of many different types of jobs the exponential distribution can be realistic for service times. Otherwise it tends to be too variable of a distribution.

Expert Solution & Answer

Answer to Problem 3.3Q

Highest in overnight hours and lowest in commuters’ hours.

Explanation of Solution

Given information:

average cars per hour(λ)=100per hourservice rate(μ) = 240 per hourno.of vehicles in the queue = Lq≤4 traffic rate(ρ)=(λμ) =100240 =0.42no: of vehicles in queue(Lq) =ρ21−ρ =0.30 ρ0=1−ρ =1−0.42 =0.58..........................2

(references from 97136-5-3.1Q)

λ=75 per hourμ=240 per hourno.of vehicles in the queue = Lq≤4ρ=( 75 240)=0.3125Lq=ρ2 1−ρ =0.142 P 0 =1−ρ =1−0.3125 =0.6875

0.103 [References from 97136-5-3.1Q]

0.58 [References from 97136-5-3.2Q]

0.6875 [References from 97136-5-3.3Q]

Hence, the probability that a vehicle finds a lane idle and gets instant service is highest for the overnight hours and lowest for the commuter hours.

Answer 6

Chapter 5, Problem 3.3Q

Summary Introduction

Interpretation: Probability of highest and lowest of a vehicle arrival.

Concept introduction:The M/M/1 queue assumes poison arrivals, exponential service times, and a single server serving customers in a FCFS fashion. Poisson arrivals are a reasonably good assumption for unscheduled systems. Further if there is a mix of many different types of jobs the exponential distribution can be realistic for service times. Otherwise it tends to be too variable of a distribution.

Expert Solution & Answer

Answer to Problem 3.3Q

Highest in overnight hours and lowest in commuters’ hours.

Explanation of Solution

Given information:

average cars per hour(λ)=100per hourservice rate(μ) = 240 per hourno.of vehicles in the queue = Lq≤4 traffic rate(ρ)=(λμ) =100240 =0.42no: of vehicles in queue(Lq) =ρ21−ρ =0.30 ρ0=1−ρ =1−0.42 =0.58..........................2

(references from 97136-5-3.1Q)

λ=75 per hourμ=240 per hourno.of vehicles in the queue = Lq≤4ρ=( 75 240)=0.3125Lq=ρ2 1−ρ =0.142 P 0 =1−ρ =1−0.3125 =0.6875

0.103 [References from 97136-5-3.1Q]

0.58 [References from 97136-5-3.2Q]

0.6875 [References from 97136-5-3.3Q]

Hence, the probability that a vehicle finds a lane idle and gets instant service is highest for the overnight hours and lowest for the commuter hours.

Answer 7

Chapter 5, Problem 3.3Q

Summary Introduction

Interpretation: Probability of highest and lowest of a vehicle arrival.

Concept introduction:The M/M/1 queue assumes poison arrivals, exponential service times, and a single server serving customers in a FCFS fashion. Poisson arrivals are a reasonably good assumption for unscheduled systems. Further if there is a mix of many different types of jobs the exponential distribution can be realistic for service times. Otherwise it tends to be too variable of a distribution.

Answer 8

Expert Solution & Answer

Answer to Problem 3.3Q

Highest in overnight hours and lowest in commuters’ hours.

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

Explanation of Solution

Given information:

average cars per hour(λ)=100per hourservice rate(μ) = 240 per hourno.of vehicles in the queue = Lq≤4 traffic rate(ρ)=(λμ) =100240 =0.42no: of vehicles in queue(Lq) =ρ21−ρ =0.30 ρ0=1−ρ =1−0.42 =0.58..........................2

(references from 97136-5-3.1Q)

λ=75 per hourμ=240 per hourno.of vehicles in the queue = Lq≤4ρ=( 75 240)=0.3125Lq=ρ2 1−ρ =0.142 P 0 =1−ρ =1−0.3125 =0.6875

0.103 [References from 97136-5-3.1Q]

0.58 [References from 97136-5-3.2Q]

0.6875 [References from 97136-5-3.3Q]

Hence, the probability that a vehicle finds a lane idle and gets instant service is highest for the overnight hours and lowest for the commuter hours.

Answer 12

Ch. 5 - Prob. 1DQ Ch. 5 - Prob. 2DQ Ch. 5 - Prob. 3DQ Ch. 5 - Prob. 4DQ Ch. 5 - Prob. 5DQ Ch. 5 - Prob. 6DQ Ch. 5 - Prob. 7DQ Ch. 5 - Prob. 1P Ch. 5 - Prob. 2P Ch. 5 - Prob. 3P

Concept explainers

Answer to Problem 3.3Q

Explanation of Solution

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Chapter 5 Solutions