QUESTION THREE (9 marks)
(a) Describe the structure and customer behavior of a queuing system
A queuing system is a framework used to model situations where customers arrive at a service facility, wait in a queue if necessary, and are served by one or more service providers. It is widely used in operations research to analyze and optimize waiting lines in systems such as banks, hospitals, call centers, and retail stores.
Structure of a Queuing System:
- Arrival Process (Input Process): Describes how customers arrive at the system. This can be deterministic or probabilistic (e.g., Poisson distribution).
- Service Mechanism (Service Process): Refers to the time it takes to serve a customer. Service times can be constant, exponentially distributed, or follow other distributions.
- Queue Discipline: The rule by which customers are selected for service. Common disciplines include First-Come, First-Served (FCFS), Last-Come, First-Served (LCFS), priority-based, and random selection.
- System Capacity: The maximum number of customers that can be accommodated in the system (waiting + being served). Can be finite or infinite.
- Number of Service Channels: Can be a single-server system or a multi-server system.
- Number of Service Stages: A system can be single-phase (one service stage) or multi-phase (sequential services).
Customer Behavior in a Queuing System:
- Balking: A customer decides not to join the queue upon arrival, usually due to long waiting lines.
- Reneging: A customer joins the queue but leaves before being served due to impatience.
- Jockeying: A customer switches between queues to reduce waiting time.
- Prioritization: Some customers are given priority based on urgency or other criteria.
(b) Arrivals at a telephone booth are considered to be Poisson, with an average time of 10 minutes between one arrival and the next. The length of a phone call is assumed to be distributed exponentially with a mean of 3 minutes.
Required: Find
Given:
- Inter-arrival time = 10 minutes → Arrival rate = 1/10 per minute = 0.1
- Mean service time = 3 minutes → Service rate = 1/3 per minute ≈ 0.333
- Utilization factor = Arrival rate / Service rate = 0.1 / 0.333 ≈ 0.3
(i) The probability that an arrival finds four persons are waiting for their turn. (2 marks)
Probability that there are exactly n customers in the system:
Probability = (1 - Utilization) × (Utilization)n
Probability that there are 4 customers in the system:
Probability = (1 - 0.3) × (0.3)4 = 0.7 × 0.0081 = 0.00567
✅ Answer: 0.00567 or 0.567%
(ii) The average number of persons waiting and making telephone calls (2 marks)
Average number of customers in the system:
Average = Utilization / (1 - Utilization) = 0.3 / 0.7 ≈ 0.4286
✅ Answer: Approximately 0.43 persons
(iii) The average length of the queue that is formed from time to time (2 marks)
Average number of customers in the queue:
Average Queue Length = (Utilization)2 / (1 - Utilization) = (0.3)2 / 0.7 = 0.09 / 0.7 ≈ 0.1286
✅ Answer: Approximately 0.13 persons
(c) The Taj service station has a central store where service mechanics arrive to take spare parts for the jobs they work upon. The mechanics wait in queue if necessary and are served on a first-come-first-served basis. The store is manned by one attendant who can attend 8 mechanics in an hour on an average. The arrival rate of the mechanic averages 6 per hour. Assuming that the pattern of mechanics arrivals is Poisson distributed and the servicing time is exponentially distributed.
Required: Determine
Given:
- Arrival rate = 6 per hour
- Service rate = 8 per hour
- Utilization = 6 / 8 = 0.75
(i) Expected time spent by a mechanic in the system (3 marks)
Expected time in the system:
Time in system = 1 / (Service rate - Arrival rate) = 1 / (8 - 6) = 1 / 2 = 0.5 hours
✅ Answer: 0.5 hours or 30 minutes
(ii) Expected time spent by a mechanic in the queue (3 marks)
Expected time in the queue:
Time in queue = (Utilization) / (Service rate - Arrival rate) = 0.75 / 2 = 0.375 hours
✅ Answer: 0.375 hours or 22.5 minutes
(iii) Expected number of mechanics in the queue (3 marks)
Expected number of mechanics in the queue:
Number in queue = Arrival rate × Time in queue = 6 × 0.375 = 2.25
✅ Answer: 2.25 mechanics