Which queuing model is represented by a situation in which customers come to a bank counter every five minutes and the banker takes exactly five minutes to serve a customer?

Correct option is D

Introduction:

A queuing model is a mathematical representation of a system where entities (customers) arrive, wait if necessary, get served, and then depart. These models are widely used in operations research, business, and computer systems to analyze service efficiency.

In this case:

• Customers arrive exactly every five minutes.

• The service time is exactly five minutes.

• There is no randomness in either the arrival time or service time.

This represents a Deterministic Queuing Model, where both arrival and service processes follow a fixed schedule without any variations or probabilities involved. key features of modal:

1. Fixed Inter-arrival Time: The time between arrivals is constant (5 minutes in this case).

2. Fixed Service Time: The service time does not vary (exactly 5 minutes).

3. No Randomness: Unlike Poisson or Exponential models, where probabilities govern arrivals and service times, a deterministic model has a predictable flow.

4. Queue Length: If the system is perfectly balanced (arrival rate = service rate), there will be no queue formation.

   
   Information Booster:
   

1. Deterministic queuing models are often used in situations where service rates and arrival rates are fixed and predictable.

2. Examples of deterministic systems:

   • Automated production lines where machines process items at a fixed rate.

   • Scheduled transport systems like trains that arrive at fixed intervals.

   • A manufacturing process where items are assembled at exactly predefined times.

3. Mathematical Representation:

   • The queuing system can be represented as D/D/1, where:

     • The first "D" represents Deterministic Arrivals (fixed intervals).

     • The second "D" represents Deterministic Service Times (fixed service time).

     • The "1" represents a single server.

4. Advantages:

   • Easy to analyze due to the absence of randomness.

   • No uncertainty in waiting times if service capacity matches demand.

     Additional Knowledge:
     

1. Poisson-exponential Single Server (a) - Incorrect:

   • The Poisson-exponential model assumes that customer arrivals follow a Poisson distribution (random arrivals), and service times follow an Exponential distribution (random service times).

   • In the given scenario, arrivals and service times are fixed, so this model does not apply.

2. Poisson-exponential Multiple Server (b) - Incorrect:

   • This model is used when multiple servers handle customer requests simultaneously, with random arrivals and random service times.

   • Since the given problem involves a single banker and fixed times, this is incorrect.

3. Poisson-exponential Dual Server (c) - Incorrect:

   • This refers to a two-server system with Poisson-distributed arrivals and exponential service times.

   • Since we have only one banker and no randomness, this is incorrect.​