Queuing System – Definition, Kendall's Notation, Performance Measures & System Stability

A queuing system is a mathematical model used to study waiting lines, where customers arrive, wait for service if necessary, 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 (line) if the server is busy, getting served, and then departing. It is used to analyze and optimize waiting times, server utilization, and overall system efficiency in real-world scenarios like banks, hospitals, and computer networks.

Basic components of a queuing system:

• Arrival process – How customers arrive (pattern/rate)
• Service mechanism – How customers are served (time/rate)
• Queue discipline – The order in which customers are served (FIFO, LIFO, Priority)
• System capacity – Maximum number of customers allowed
• Number of servers – Single or multiple servers

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Kendall's Notation

Kendall's notation is a standard shorthand used to describe and classify a queuing system. It is written as:

$$A / B / s / K / N / D$$

Where each symbol represents:

Position	Symbol	Meaning
A	Arrival process	Distribution of inter-arrival times
B	Service process	Distribution of service times
s	Number of servers	Count of parallel servers
K	System capacity	Maximum customers in system (queue + service)
N	Population size	Size of the calling population
D	Queue discipline	Rule for selecting next customer

Common distribution codes:

• M – Markovian (Exponential/Poisson – memoryless)
• D – Deterministic (constant/fixed times)
• G – General (any arbitrary distribution)
• Ek – Erlang-k distribution

Example: $M/M/1/\infty/\infty/FIFO$ represents a single server system with Poisson arrivals, exponential service times, infinite capacity, infinite population, and First-In-First-Out discipline.

When last three parameters are not mentioned (e.g., M/M/1$), it is assumed: $K = \infty, $N = \infty$, $D = FIFO$.

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Performance Measures in Single Server Queuing System (M/M/1)

Let:

• $\lambda$ = mean arrival rate
• $\mu$ = mean service rate
• $\rho = \lambda / \mu$ = traffic intensity (utilization factor)

The key performance measures are:

Measure	Formula	Meaning
Traffic Intensity ($\rho$)	$\rho = \lambda / \mu$	Fraction of time server is busy
Average number in system ($L_s$)	$L_s = \frac{\rho}{1 - \rho}$	Expected customers in entire system
Average number in queue ($L_q$)	$L_q = \frac{\rho^2}{1 - \rho}$	Expected customers waiting in queue
Average time in system ($W_s$)	$W_s = \frac{1}{\mu - \lambda}$	Expected time a customer spends in system
Average time in queue ($W_q$)	$W_q = \frac{\rho}{\mu - \lambda}$	Expected waiting time in queue only
Probability system is empty ($P_0$)	$P_0 = 1 - \rho$	Probability of zero customers
Probability of n customers ($P_n$)	$P_n = (1-\rho)\rho^n$	Probability of exactly n customers

Little's Law connects these measures:

$$L_s = \lambda \cdot W_s \quad \text{and} \quad L_q = \lambda \cdot W_q$$

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System Stability – Which Measure Determines It and How?

The traffic intensity $\rho = \lambda / \mu$ is the critical measure that determines system stability.

Stability condition: The system is stable (steady-state exists) if and only if:

$$\rho = \frac{\lambda}{\mu} < 1$$

Explanation:

• If $\rho < 1$: The service rate is greater than the arrival rate ($\mu > \lambda$). The server can handle incoming customers, queue does not grow indefinitely, and the system reaches a steady state.

• If $\rho = 1$: The arrival rate equals the service rate. The queue grows without bound over time — system is unstable.

• If $\rho > 1$: Customers arrive faster than they are served. The queue length increases infinitely — system is completely unstable and no steady-state solution exists.

Conclusion: The traffic intensity $\rho$ acts as the stability indicator. For any single server queuing system to function effectively and reach equilibrium, it is mandatory that $\rho < 1$, ensuring that the server is not overwhelmed and all performance measures remain finite and meaningful.