Suppose that customer interarrival times at a quick serve restaurant are exponentially distributed with an arrival rate of 5 customers per hour. Suppose the average customer service time is 9 minutes and service times exponentially distributed.

The goal is to simulate the arrival of 200 customers with the first customer arriving at time 0. Assume that the units for the simulation are in minutes.

The spreadsheet below was provided with the quiz.

Assume that the correct formulas are entered for the simulation including the formulas for the actual simulation times and the theoretical times, provided by the M/M/1 queuing model, in cells F3 thorough G4. Suppose, to better capture steady state performance, these formulas are based only on the last 100 customers in the 200 customer simulation. All summary times are in minutes.

The data table (I7-K307) contains 300 runs of the simulation with summary statistics displayed above. Both the time in system and time in queue statistics are captured in minutes.

Which of the following statements are true?

Question

Suppose that customer interarrival times at a quick serve restaurant are exponentially distributed with an arrival rate of 5 customers per hour.   Suppose the average customer service time is 9 minutes and service times exponentially distributed.

The goal is to simulate the arrival of 200 customers with the first customer arriving at time 0.   Assume that the units for the simulation are in minutes.

The spreadsheet below was provided with the quiz.

Assume that the correct formulas are entered for the simulation including the formulas for the actual simulation times and the theoretical times, provided by the M/M/1 queuing model, in cells F3 thorough G4.  Suppose, to better capture steady state performance, these formulas are based only on the last 100 customers in the 200 customer simulation.  All summary times are in minutes.

The data table (I7-K307) contains 300 runs of the simulation with summary statistics displayed above.  Both the time in system and time in queue statistics are captured in minutes.

Which of the following statements are true?

BlackTom AI · Accepted Answer

To tackle this question, I’ll examine how each statement aligns with standard M/M/1 queueing behavior and basic queueing intuition, using the given system parameters (arrival rate λ = 5 customers/hour, service rate μ = 1 / (9 minutes) = 6.666... customers/hour, and a stable system since ρ = λ/μ = 0.75).

Option A: If the arrival rate is increased, and no other changes are made, then, on average the time in queue and system will increase.
- Reasoning: In an M/M/1 queue, increasing the arrival rate while keeping the service rate fixed increases utilization ρ. Higher utilization generally leads to longer waits in the queue and longer time in the system, because customers spend more time awaiting service and more customers accumulate in the system. This statement aligns with queueing theory and is true under the usual assumptions (Poisson arrivals, exponential service, single server).

Option B: If the service time is increased, and no other changes are made, then, on average the time in queue and system will decrease.
- Reasoning: Increasing service time reduces the service rate μ. A slower service rate means customers are served more slowly, causing more accumulation and longer waiting times. Therefore both time in queue and time in system should increase, not decrease. This statement is false.

Option C: If the time between arrivals is increased, and no other changes are made, then, on average the time in queue and system will decrease.
- Reasoning: Increasing the time between arrivals is equivalent to reducing the arrival rate λ. With a fixed μ, a lower λ lowers utilization ρ, reducing congestion. In M/M/1 terms, reduced arrival rate tends to shorten both the queue and the total time in the system. This statement is true.

Option D: If the service rate is increased, and no other changes are made, then, on average the time in queue and system will increase.
- Reasoning: Increasing the service rate μ (faster service) increases system throughput and reduces queue buildup, thereby decreasing both time in the queue and time in the system. This statement is false.

Option E: Exactly two answers are correct.
- Reasoning: At first glance, A and C appear correct based on standard M/M/1 behavior. However, the meta-claim here hinges on how many options are truly correct when considering the definitions and potential edge interactions (e.g., the relationship with ρ and whether any other option could become correct under those same conditions). If A and C are the only correct statements, then the assertion “Exactly two answers are correct” would be true. But if E itself is true, it would imply there are exactly two correct statements, which would force E to be true while also affecting the count of true statements in a self-referential way. In common multiple-choice design, such a self-referential option often serves as a trap; the practical interpretation is that A and C are the core correct statements, and E adds a circular meta-claim that depends on the count of correct options. In typical test construction, we would treat E as a statement that is not independently verifiable without committing to which other statements are correct, making its truth value ambiguous in isolation. This makes E a problematic choice if you’re aiming for a clean, non-paradoxical answer set.

Option F: Exactly three answers are correct.
- Reasoning: If A and C are correct, and E is considered true only if there are exactly two correct statements, then having three correct statements would render F false. Conversely, if some other option were also true, you would need a third correct option, which would alter the truth of E and F in a way that creates an inconsistent self-reference. Practically, under standard queueing intuition, only A and C are robustly correct, while B and D are clearly false; E and F become logically entangled and do not provide a consistent independent truth value in this context. Therefore this option is not the best standalone truth assertion.

Summary perspective: The most solid, conceptually consistent conclusions are that increasing arrival rate (A) tends to increase times, and increasing the time between arrivals (C) tends to reduce times. B and D are contrary to queueing intuition and are false. Because of the self-referential nature of E and F in relation to the count of correct statements, those options are not straightforwardly true in a way that yields a clean, non-contradictory selection. Therefore, the statements you should identify as true are A and C, with E and F creating inconsistent meta-statements about the number of correct choices.

MGMT 101 LEC ONL: MANA... Study Checkpoint 7: Analysis of Queues

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