TRB PG Assistant - Queuing theory - single server and multi-server models, Erlang service distributions, cost models and optimization Question Bank with Options

1. What is the main challenge in applying queuing theory to real-world problems?

A) The arrival and service processes are always perfectly exponential

B) Obtaining accurate estimates for arrival and service rates, and ensuring model assumptions hold

C) Queuing models always predict infinite queues

D) The cost of implementing queuing solutions is always prohibitive

2. If a system is unstable (e.g., λ > μ in M/M/1), what is the expected behavior of the queue?

A) It will eventually clear

B) It will grow indefinitely

C) It will fluctuate around a constant length

D) It will reach a steady state quickly

3. Which of the following statements about the utilization factor (ρ) in an M/M/c system is true?

A) ρ = λ / μ

B) ρ = λ / (cμ)

C) ρ = cμ / λ

D) ρ = μ / λ

4. The Erlang distribution is a special case of the Gamma distribution. What is its primary application in queuing theory?

A) Modeling arrival processes

B) Modeling queue discipline

C) Modeling service times when they are not exponential but have a specific variance

D) Modeling server availability

5. What is the impact of reducing the service time (increasing μ) in a stable M/M/1 system?

A) Increases L, increases Wq

B) Decreases L, decreases Wq

C) Increases L, decreases Wq

D) Decreases L, increases Wq

6. Which of the following decisions can be informed by queuing theory analysis?

A) Product pricing

B) Advertising strategy

C) Number of service personnel to employ

D) Inventory management

7. In multi-server systems, if the number of servers 'c' is very large, the system behavior approaches that of:

A) A single-server system

B) An infinite-server system

C) A system with no queue

D) A finite-capacity system

8. What is the primary characteristic of an M/D/1 queue (constant service time)?

A) Exponential arrivals, exponential service, 1 server

B) Exponential arrivals, constant service, 1 server

C) Constant arrivals, exponential service, 1 server

D) Constant arrivals, constant service, 1 server

9. The term 'traffic intensity' in a queuing system is often synonymous with:

A) Arrival rate

B) Service rate

C) Utilization factor (ρ)

D) Queue length

10. If a queuing system has a finite queue capacity, what is a potential consequence?

A) Infinite waiting time

B) Increased server utilization

C) Lost customers due to blockage

D) Decreased arrival rate

11. What is the probability of zero customers in an M/M/1 system where ρ = λ/μ?

A) ρ

B) 1 - ρ

C) 1 + ρ

D) ρ^2

12. Which scenario would typically NOT be modeled using basic M/M/1 or M/M/c models?

A) A single checkout counter at a small store

B) Multiple tellers at a bank

C) A call center with many operators

D) A system with non-exponential service times and batch arrivals

13. In the context of Erlang distributions, what does the parameter 'k' (number of phases) represent?

A) The arrival rate

B) The service rate

C) The number of sequential exponential stages the service time passes through

D) The number of servers

14. The Little's Law states that:

A) L = λW and Lq = λWq

B) L = μW and Lq = μWq

C) W = λL and Wq = λLq

D) W = μL and Wq = μLq

15. What is the steady-state condition in queuing theory?

A) The system is always empty

B) The probability distributions of system state variables do not change over time

C) The arrival rate equals the service rate

D) The queue length is always zero

16. In a cost model, if the cost of waiting is very high compared to the cost of service, what strategy would likely be chosen?

A) Reduce the number of servers

B) Increase the service rate

C) Increase the number of servers or service rate

D) Increase the arrival rate

17. What does 'queue capacity' refer to?

A) The maximum number of customers that can be served simultaneously

B) The maximum number of customers allowed in the system

C) The maximum number of customers allowed to wait in the queue

D) The rate at which customers are served

18. Which of the following is a characteristic of a Poisson arrival process?

A) The number of arrivals in non-overlapping time intervals are dependent

B) The probability of an arrival in a small interval is proportional to the length of the interval

C) The inter-arrival times are constant

D) The service times are constant

19. If the arrival rate (λ) increases while the service rate (μ) and number of servers (c) remain constant, what happens to the average waiting time?

A) It decreases

B) It stays the same

C) It increases

D) It becomes zero

20. The Erlang-B formula is used to calculate:

A) The probability of waiting in a multi-server system

B) The probability of all servers being busy (blocking) in a multi-server system with no queue

C) The average queue length

D) The utilization factor

21. What does the 'G' in Kendall notation (e.g., M/G/1) represent?

A) General distribution of service times

B) Geometric distribution of arrivals

C) Gaussian distribution

D) Group arrivals

22. Which of the following is NOT a component of a queuing system definition?

A) Arrival process

B) Service process

C) Queue discipline

D) Customer balking

23. In an M/M/c system, what is 'c'?

A) The number of customers in the queue

B) The service rate per server

C) The number of parallel servers

D) The arrival rate

24. What is the average time a customer spends in the system (W) for an M/M/1 queue?

A) W = Wq + 1/μ

B) W = Wq - 1/μ

C) W = Lq / λ

D) W = L / μ

25. If λ = 5 customers/hour and μ = 10 customers/hour in an M/M/1 system, what is the average number of customers in the queue (Lq)?

A) 0.25

B) 0.5

C) 1.0

D) 2.0

26. The Kendall notation 'M/M/1' describes a queuing system with:

A) Exponential arrivals, exponential service, and 1 server

B) Constant arrivals, constant service, and 1 server

C) Poisson arrivals, exponential service, and 1 server

D) Exponential arrivals, Poisson service, and 1 server

27. What does the 'M' in M/M/1 notation signify regarding arrival and service processes?

A) Markovian

B) Maximum

C) Minimum

D) Median

28. Which distribution assumes that the time between arrivals follows an exponential distribution?

A) Erlang distribution

B) Poisson distribution

C) Normal distribution

D) Uniform distribution

29. Optimization in queuing theory aims to find the point where:

A) Waiting time is minimized, regardless of cost

B) Service cost is minimized, regardless of waiting time

C) Total cost is minimized

D) The number of servers is maximized

30. The total cost in a queuing system is generally the sum of:

A) Waiting cost and service cost

B) Arrival cost and waiting cost

C) Service cost and balking cost

D) Server cost and queue cost

31. Which cost model in queuing theory considers the cost of providing service?

A) Waiting cost

B) Service cost

C) System cost

D) Customer cost

32. What does 'reneging' mean in queuing theory?

A) A customer deciding not to join the queue because it is too long

B) A server being busy

C) A customer leaving the queue before being served

D) A server closing down temporarily

33. What does 'balking' mean in queuing theory?

A) A customer leaving the queue before being served

B) A customer deciding not to join the queue because it is too long

C) A server closing down temporarily

D) A server being busy

34. In a multi-server system (M/M/c), what is the condition for stability?

A) cμ > λ

B) λ > cμ

C) cλ > μ

D) μ > cλ

35. What is the key difference between the M/M/1 and M/M/c models?

A) Number of servers

B) Arrival distribution

C) Service distribution

D) Queue discipline

36. Erlang service distributions are characterized by:

A) A constant service time

B) A single parameter, the mean service time

C) Two parameters: the number of phases and the mean service time per phase

D) Exponential service times

37. What is the 'Erlang C formula' used for?

A) Calculating average queue length in M/M/1

B) Calculating the probability of a customer having to wait in a multi-server system

C) Determining the optimal number of servers

D) Calculating the service rate

38. In an M/M/1 queue, what is the average waiting time in the queue (Wq)?

A) λ / (μ * (μ - λ))

B) μ / (λ * (μ - λ))

C) 1 / (μ - λ)

D) λ / (μ * (λ - μ))

39. For an M/M/1 queue, what is the average number of customers in the system (L)?

A) λ / (μ - λ)

B) μ / (μ - λ)

C) λ / (λ - μ)

D) μ / (λ - μ)

40. What is the utilization factor (ρ) of a single-server system with arrival rate λ and service rate μ?

A) μ / λ

B) λ / μ

C) (λ + μ) / λ

D) (λ + μ) / μ

41. The condition for a stable queuing system (M/M/1) is:

A) λ > μ

B) μ > λ

C) λ = μ

D) λ * μ > 1

42. In a multi-server model, what is 'c' usually denoted as?

A) The arrival rate

B) The service rate per server

C) The number of parallel servers

D) The queue capacity

43. Which queuing model has only one service channel?

A) Multi-server model

B) Single-server model

C) Infinite server model

D) Finite server model

44. What does 'FIFO' stand for in the context of queue discipline?

A) Fast In, Fast Out

B) First In, First Out

C) Free In, Free Out

D) Fixed In, Fixed Out

45. Which of the following is a common assumption in basic queuing models?

A) Infinite queue capacity

B) Finite number of customers

C) Exponential service times

D) No balking or reneging

46. What does the symbol 'μ' typically represent in a queuing system?

A) The arrival rate

B) The service rate

C) The number of servers

D) The queue length

47. What does the symbol 'λ' typically represent in a queuing system?

A) The service rate

B) The average number of customers in the system

C) The arrival rate

D) The average waiting time

48. In queuing theory, what does 'server' refer to?

A) A customer waiting in the queue

B) The arrival rate of customers

C) A facility or person providing service

D) The queue discipline

49. What is the primary goal of Queuing Theory?

A) To maximize waiting time for customers

B) To minimize the number of servers

C) To balance service cost and waiting cost

D) To eliminate all queues instantaneously

Queuing theory - single server and multi-server models, Erlang service distributions, cost models and optimization - Question Bank