Queuing System: Definition, Disciplines, and Performance Measures

A queuing system is a mathematical model used to study waiting lines where customers arrive, wait for service if it is not immediately available, and leave after being served.

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Definition of Queuing System

A queuing system consists of customers arriving at a service facility, waiting in a queue (if necessary), getting served, and then departing. It is represented using Kendall's notation: $A/B/c/K/N/D$, where A = arrival distribution, B = service distribution, c = number of servers, K = system capacity, N = population size, D = queue discipline.

Key components of a queuing system:

• Arrival process – Pattern in which customers arrive (e.g., Poisson process)
• Service mechanism – How customers are served (number of servers, service time distribution)
• Queue discipline – Rule that determines the order of service
• System capacity – Maximum number of customers allowed in the system
• Customer population – Finite or infinite source of customers

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Queuing Disciplines

Queue discipline defines the rule by which customers are selected from the queue for service.

• FCFS (First Come, First Served) – The customer who arrives first is served first. This is the most common discipline (e.g., bank queues, ticket counters).

• LCFS (Last Come, First Served) – The customer who arrives last is served first. This works like a stack (e.g., elevator system, stack of plates).

• SIRO (Service In Random Order) – Customers are selected randomly from the queue regardless of their arrival time (e.g., lottery-based token systems).

• Priority Scheduling – Customers are served based on assigned priority levels rather than arrival order.

  • Preemptive Priority – A higher priority customer can interrupt the service of a lower priority customer.
  • Non-Preemptive Priority – The current service is completed before serving the higher priority customer.

• Round Robin – Each customer gets a fixed time slice of service in a cyclic manner (e.g., CPU scheduling in operating systems).

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Performance Measures for Evaluation of Queuing System

These are the metrics used to evaluate how efficiently a queuing system operates:

Symbol	Performance Measure	Meaning
$L_s$	Average number of customers in the system	Total customers (in queue + being served)
$L_q$	Average number of customers in the queue	Customers waiting only
$W_s$	Average time a customer spends in the system	Total time (waiting + service)
$W_q$	Average waiting time in the queue	Time spent waiting only
$\rho$	Server utilization (traffic intensity)	Fraction of time server is busy
$P_0$	Probability of system being empty	No customer in the system

Key formulas (for M/M/1 queue):

• Traffic intensity: $\rho = \frac{\lambda}{\mu}$, where $\lambda$ = arrival rate, $\mu$ = service rate

• Average number in system: $L_s = \frac{\lambda}{\mu - \lambda}$

• Average number in queue: $L_q = \frac{\lambda^2}{\mu(\mu - \lambda)}$

• Average time in system: $W_s = \frac{1}{\mu - \lambda}$

• Average waiting time in queue: $W_q = \frac{\lambda}{\mu(\mu - \lambda)}$

• Little's Formula: $L_s = \lambda \cdot W_s$ and $L_q = \lambda \cdot W_q$

• Server utilization: $\rho = \frac{\lambda}{\mu}$ (system is stable only when $\rho < 1$)

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Conclusion

A queuing system provides a mathematical framework to analyze waiting lines. By understanding different queue disciplines and measuring performance parameters like $L_s$, $W_q$, and $\rho$, we can optimize service systems to reduce waiting time and improve efficiency in real-world applications like hospitals, banks, and computer networks.