• These notes cover the introduction to queueing systems and models.

Basic Structure of Queueing Models

• A queueing system involves an input source(calling population) which can be finite or infinite.
• Customers enter a queue.
• They are then served by a service mechanism.
• Finally exit the system as served customers.
• Interarrival time (IAT) is the time between consecutive customer arrivals.
• "Random" arrivals with a known average rate are often modeled as a Poisson process with a mean arrival rate denoted as λ.
• The queue has a maximum permissible number of customers, which may be infinite or finite.
• Queueing systems are common in traffic lights, bank tellers, airport security, emergency rooms, toll booths, parking lots, and call centers.
• The size of the queue (either finite or infinite) affects the performance of the queueing system with behaviors like balking or reneging.
• The service discipline such as FIFO, LIFO, priority, or random affects the queueing system's performance.
• Service Discipline is the order in which customers are selected for service and include FIFO, LIFO, priority, and random.
• Service Mechanism refers to one or more service facilities and can have one or more parallel service channels, also known as servers.
• Service Time must provide the distribution of service times and parameters such as constant or exponential.
• Assuming a Poisson arrival process implies that interarrival times between customers are independent and exponentially distributed which simplifies the analysis.
• Increasing the number of servers in the service mechanism reduces waiting time and queue length, this must be balanced against additional costs, and the number of servers depends on the trade-off between cost and performance.

Examples of Real Queueing Systems

• Commercial service systems include barber shops, bank teller services, supermarket checkouts, and gas stations.
• Transportation service systems include toll booths, traffic lights, truck or ship loading/unloading areas, parking lots (where spaces act as servers), taxis, and elevators.
• Industrial service systems cover maintenance repair crews, inspection stations, and secretarial/clerical pools.
• Social service systems include judicial and legislative systems (processing bills), and healthcare systems.

Terminology

• State of the System is the number of customers in the queueing system, including those being served and those waiting.
• Queue Length is the number of customers waiting for service. This can be calculated as the State of the System less the number of customers being served.
• λ (lambda) is the mean arrival rate and denotes the expected interarrival time.
• µ (mu) represents the mean service rate and denotes the expected service time.
• ρ (rho) is the utilization factor and is the expected fraction of time the servers are busy.
• In a bank teller service, the state of the system equals the number of customers in the bank which includes those being served and those waiting in line.
• In a parking lot queueing system, the number of cars in the parking lot is the state of the system and the cars waiting for an available space determines the queue length.

Steady-State Condition

• After enough time, the state of the system becomes independent of its initial state and the time elapsed.
• The system has reached a state of dynamic equilibrium when the rate in equals the rate out.
• For dynamic equilibrium to occur, the utilization factor must be less than 1 (ρ < 1).
• The utilization factor (ρ) is the expected fraction of time servers are busy.
• If ρ ≥ 1, it means the arrival rate is greater than or equal to the service rate, which leads to an unstable system with queue length growing without bound.
• If the steady-state condition is not met, then the system may not reach a predictable or stable state and performance measures are heavily influenced by initial conditions or time-dependent factors.
• Changes in arrival rate, service rate, or the number of servers can affect the steady-state condition and performance, requiring the system to adapt to maintain dynamic equilibrium.
• A fast-food restaurant can experience significant customer arrival variations during lunch and dinner hours which can make maintaining the steady-state condition challenging.
• A hospital emergency room will experience sudden patient influxes disrupting steady-state and requires quick adaptation to maintain performance.

Measures of Performance at Steady-State

• L = Expected number of customers in the queueing system
• Lq = Expected number of customers in the queue
• W = Expected waiting time in the system, including service time
• Wq = Expected waiting time in the queue
• For a bank teller, expected waiting time in queue (Wq) affects customer satisfaction and is an important performance measure.
• In a hospital emergency room, the expected number of patients in the queue (Lq) would be a key measure for adjusting staffing and indicating potential long wait times.

Classifying Queueing Systems

• Number of Phases: How many workstations a customer might visit (service facility).
• Number of Channels: How many servers at a workstation.
• Queue discipline: Rules for determining which customer in the waiting line is served next, such as FIFO, LIFO, random, or priority.
  • Queue discipline can have preemptive priority, where a higher priority entity can interrupt the service of a lower priority one.
• Other considerations include behaviors like balking, reneging, and jockeying.

The P/M/M/S/Q Queue

• Queueing models may be denoted by p/M/M/s/q.
• A bank teller service could be modeled as an M/M/s/q queue, where customer arrivals follow a Poisson process, service times are exponential, and there is a finite number of tellers and queue capacity.
• A call center can be represented as an M/M/s/q queue, where incoming calls follow a Poisson arrival process, service times are exponentially distributed, and there is a limited number of representatives with a finite queue capacity.

The Queueing Model M/M/1

• A single-phase, single-server, FIFO queueing system.
• In this model:
  • Infinite source (p = ∞)
  • Poisson arrivals
  • Exponential service times
  • One server (s=1)
  • Infinite queue space (q = ∞)
• M/M/1 queueing model
• Single-phase
• Single-server
• FIFO queueing system
• Poisson arrivals
• Exponential service times
• Infinite queue capacity.
• Key assumptions are that the arrivals follow a Poisson process, the service times are exponentially distributed, and there is a single server; the accuracy of the model is limited by these assumptions as they may not always hold true.

The M/M/S Queue

• A single-phase, multiple-server, FIFO queueing system.
• This model contains the same properties of the M/M/1 (Infinite source, Poisson arrivals, Exponential service times) but with:

  • servers > 1

  • Infinite queue space

Steady-State Measures of Performance for The M/M/S Queue

• P = probability that there are n customers in the system n
• P = probability that there are no customers in the system = 0 Po
• Lq = Expected number of customers in queue
• Wq = Expected waiting time in the queue
• L = Lq + p (Expected number of customers in the system = Expected number of customers in queue plus utilization factor)
• W = Wq + (Expected waiting time in system = Expected waiting time in queue plus some additional time)
• To compute P use other equations depending on if n ≤ s, P, if n > s, P = use other equations as well
• Relationships among measures of performance
  • L = λ W (Expected number of customers in the system = Mean arrival rate times Expected waiting time in the system)
  • Lq = λ Wq (Expected number of customers in the queue = Mean arrival rate times Expected length of the queue)
  • W = Wq + (Expected waiting time in system = Expected waiting time in queue )

Conclusion

• Queueing models are often subjected to simulation for various reasons.
• This elements are used repeatedly in simulation courses.