Match List I with List II List I List II A. Markovian property I. Rule of determining the order in which members of the queue are selected to begin service B. Waiting time in system II. The time of next arrival is completely uninfluenced by when the last arrival occurred C. Steady state condition III. The queueing is in after operating for some time with a fixed utilisation factor less than one D. Queue discipline IV. Elapsed time that an individual customer spends in queue, both before service and during service Choose the correct answer from the options given below:
Understanding Key Concepts in Queueing Theory
Queueing theory is a mathematical study of waiting lines, or queues. It helps in analysing processes that involve waiting, such as customers waiting for service at a bank or jobs waiting for processing on a computer. Understanding the basic terminology is crucial for analysing queueing systems. Let's examine the concepts given in List I and match them with their appropriate descriptions in List II.
Analysing the Queueing Theory Terms and Definitions
Let's break down each term from List I and find its corresponding definition or description in List II:
- Markovian property: This property is characteristic of processes where the future state depends only on the current state, not on the sequence of events that preceded it. In queueing theory, it often relates to arrival times and service times following an exponential distribution, which is memoryless. This means the probability of an arrival or service completion in the next small interval of time does not depend on how long it has been since the last event. List II, point II, describes this memoryless nature specifically for arrivals: "The time of next arrival is completely uninfluenced by when the last arrival occurred". This aligns perfectly with the Markovian property when applied to arrival processes.
- Waiting time in system: This refers to the total time a customer or item spends within the queueing system. It includes the time spent waiting in the queue before service begins plus the time spent receiving the service. List II, point IV, states "Elapsed time that an individual customer spends in queue, both before service and during service". This definition precisely matches the concept of waiting time in the system.
- Steady state condition: A queueing system reaches a steady state when it has been operating for a sufficiently long time and its statistical properties (like average queue length or average waiting time) become stable and independent of the initial conditions. This state is typically reached when the system's utilization factor (arrival rate divided by service rate) is less than one. List II, point III, describes this: "The queueing is in after operating for some time with a fixed utilisation factor less than one". This accurately captures the condition for a queueing system to reach steady state.
- Queue discipline: This is the rule or policy that determines the order in which customers waiting in the queue are selected for service. Common queue disciplines include First-In, First-Out (FIFO) or First-Come, First-Served (FCFS), Last-In, First-Out (LIFO), Service In Random Order (SIRO), and Priority. List II, point I, defines this as: "Rule of determining the order in which members of the queue are selected to begin service". This is the exact definition of queue discipline.
Matching List I with List II
Based on the analysis above, we can create the following matches:
| List I (Concept) | List II (Description/Definition) | Match |
|---|---|---|
| A. Markovian property | II. The time of next arrival is completely uninfluenced by when the last arrival occurred | A - II |
| B. Waiting time in system | IV. Elapsed time that an individual customer spends in queue, both before service and during service | B - IV |
| C. Steady state condition | III. The queueing is in after operating for some time with a fixed utilisation factor less than one | C - III |
| D. Queue discipline | I. Rule of determining the order in which members of the queue are selected to begin service | D - I |
The correct matching is A - II, B - IV, C - III, D - I.
Let's look at the provided options and find the one that corresponds to these matches.
- Option 1: A - II, B - I, C - III, D - IV (Incorrect match for B and D)
- Option 2: A - II, B - I, C - III, D - IV (Same as Option 1, incorrect)
- Option 3: A - II, B - IV, C - III, D - I (Correct matches)
- Option 4: A - III, B - IV, C - II, D - I (Incorrect match for A and C)
Therefore, the option that correctly matches the concepts in List I with their descriptions in List II is Option 3.
Revision Table: Queueing Theory Concepts
Review the key terms and their definitions to reinforce your understanding of queueing theory basics.
| Concept | Definition/Description | Related Idea |
|---|---|---|
| Markovian property | Memoryless property; future state depends only on current state. Often applies to exponential arrival/service times. | Exponential distribution |
| Waiting time in system | Total time spent in the system (queue time & service time). | Total time in system |
| Steady state condition | Long-run equilibrium of the system when utilization < 1. | System stability |
| Queue discipline | Rule for selecting the next customer for service (e.g., FIFO, LIFO). | Service order |
Additional Information: Further Queueing Analysis
Beyond these basic concepts, queueing theory involves analyzing various aspects of waiting lines, including:
- Arrival Process: How customers arrive (e.g., Poisson arrival process, which exhibits the Markovian property).
- Service Process: How long service takes (e.g., Exponential service times, also Markovian; constant service times; general distributions).
- Number of Servers: How many service channels are available.
- System Capacity: The maximum number of customers allowed in the system (queue + service).
- Population Size: Whether customers come from a finite or infinite pool.
These factors, combined with the queue discipline, determine key performance measures like average queue length, average waiting time in queue, average waiting time in system, and system utilization.
Important Questions from Operations Research
A transportation problem with m origins and n destinations becomes a trans-shipment problem with
Customers arrive at a reception counter at an average interval rate of 10 minutes and the receptionist takes an average of 6 minutes for one customer. The average queue length is:
The first and foremost important feature for a project to be successful is:
In Queuing Theory, statistical pattern by which customers arrive over a period of time, follows
Arrange the operations in production planning and control in correct sequence :
(i) Routing
(ii) Dispatching
(iii) Follow up
(iv) Scheduling
Choose the correct answer from the code given below:
