Glossary

21.2 Single-server and multi-server queues

3 min read•july 19, 2024

Queues are everywhere, from bank lines to . Single-server queues have one server, while multi-server queues have multiple. Both types use measures like and to gauge performance.

M/M/1 and M/M/c models help analyze these queues. They assume Poisson arrivals and exponential service times. These models let us calculate important stats like average wait times and system utilization, which are key for efficient operations.

Single-server Queues

Performance measures of single-server queues

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  • Average waiting time (WqW_q) represents the mean duration a customer spends in the queue before being served (bank teller line)
  • Average queue length (LqL_q) indicates the mean number of customers waiting in the queue at any given time (supermarket checkout)
  • Server utilization (ρ\rho) measures the proportion of time the server is occupied serving customers (call center agent)
  • establishes a relationship between the average queue length (LqL_q), average waiting time (WqW_q), and (λ\lambda) as Lq=λWqL_q = \lambda W_q
  • condition requires the arrival rate to be less than the (λ<μ\lambda < \mu) to prevent the queue from growing indefinitely (stable system)

Applications of M/M/1 queuing model

  • M/M/1 queuing model assumes Poisson arrival process with rate λ\lambda, exponential service times with rate μ\mu, a single server, infinite queue capacity, and first-come, first-served (FCFS) discipline (fast food drive-thru)
  • Server utilization formula: ρ=λμ\rho = \frac{\lambda}{\mu}
  • Average number of customers in the system: L=ρ1−ρL = \frac{\rho}{1-\rho}
  • Average number of customers in the queue: Lq=ρ21−ρL_q = \frac{\rho^2}{1-\rho}
  • Average time spent in the system: W=1μ−λW = \frac{1}{\mu-\lambda}
  • Average waiting time in the queue: Wq=ρμ−λW_q = \frac{\rho}{\mu-\lambda}
  • Probability of having nn customers in the system: Pn=(1−ρ)ρnP_n = (1-\rho)\rho^n (used to determine staffing requirements)

Multi-server Queues

Performance measures of multi-server queues

  • Average waiting time in the queue (WqW_q) represents the mean duration a customer spends waiting before being served by one of the multiple servers (airline check-in counters)
  • Average queue length (LqL_q) indicates the mean number of customers waiting in the queue for service from any of the servers (hospital emergency room)
  • System utilization (ρ\rho) measures the proportion of time all servers are simultaneously busy (call center with multiple agents)
  • Multi-server queues typically exhibit shorter waiting times and queue lengths compared to single-server queues with identical arrival and service rates (increased efficiency)

Applications of M/M/c queuing model

  • M/M/c queuing model assumes Poisson arrival process with rate λ\lambda, exponential service times with rate μ\mu, cc servers, infinite queue capacity, and FCFS discipline (bank with multiple tellers)
  • System utilization formula: ρ=λcμ\rho = \frac{\lambda}{c\mu}
  • Probability of having 0 customers in the system: P0=[∑n=0c−1(cρ)nn!+(cρ)cc!(1−ρ)]−1P_0 = \left[\sum_{n=0}^{c-1}\frac{(c\rho)^n}{n!}+\frac{(c\rho)^c}{c!(1-\rho)}\right]^{-1}
  • Average number of customers in the queue: Lq=(cρ)cρc!(1−ρ)2P0L_q = \frac{(c\rho)^c\rho}{c!(1-\rho)^2}P_0
  • Average waiting time in the queue: Wq=LqλW_q = \frac{L_q}{\lambda}
  • Average number of customers in the system: L=Lq+cρL = L_q + c\rho
  • Average time spent in the system: W=Wq+1μW = W_q + \frac{1}{\mu} (used to optimize resource allocation and minimize customer wait times)

Key Terms to Review (18)

Arrival rate: The arrival rate refers to the frequency at which entities, such as customers or requests, arrive at a service point over a specified time period. This rate is critical in queuing theory as it helps determine how busy a system will be and affects both waiting times and system performance. Understanding the arrival rate is essential for modeling both single-server and multi-server queues, as it directly influences the overall efficiency of service operations.
Balking: Balking refers to the behavior of potential customers who, upon seeing a queue, decide not to join it and leave the waiting area instead. This phenomenon is crucial in the analysis of single-server and multi-server queues, as it impacts the overall customer flow and service efficiency. Understanding balking helps in designing systems that minimize waiting times and optimize service processes, ultimately leading to improved customer satisfaction and operational effectiveness.
Call Centers: Call centers are centralized offices used for receiving or transmitting a large volume of inquiries by telephone, often serving as the primary point of contact for customers. They can operate as single-server systems, where one agent handles customer calls, or multi-server systems, where multiple agents work simultaneously to manage higher volumes of calls, allowing for improved efficiency and reduced wait times for customers.
Erlang B Formula: The Erlang B formula is a mathematical equation used in telecommunications to calculate the probability of call blocking in a system with a fixed number of servers and Poisson distributed call arrivals. It helps in determining how many resources (like phone lines) are needed to maintain a desired level of service by quantifying the relationship between traffic intensity and call blocking probability. The formula is critical for designing systems that manage incoming calls efficiently, whether they are single-server or multi-server configurations.
Exponential Distribution: The exponential distribution is a continuous probability distribution often used to model the time until an event occurs, such as the time until a radioactive particle decays or the time until the next customer arrives at a service point. It is characterized by its constant hazard rate and memoryless property, making it closely related to processes like queuing and reliability analysis.
Little's Law: Little's Law is a fundamental theorem in queuing theory that describes the relationship between the average number of items in a system, the average arrival rate of items, and the average time an item spends in the system. This law establishes a clear connection between these variables, stating that the average number of items in a queuing system is equal to the product of the arrival rate and the average time spent in the system. Understanding this relationship is crucial in analyzing various processes, including birth-death processes and both single-server and multi-server queues.
M/m/1 model: The m/m/1 model is a basic queueing model used in operations research and telecommunications, characterized by a single server serving a queue of customers arriving at a random rate, following a Poisson process, and being served at an exponential service rate. This model is important for understanding the behavior of systems where customers wait in line for service, and it highlights the relationship between arrival rates, service rates, and system performance metrics such as waiting times and queue lengths.
M/m/c model: The m/m/c model is a mathematical representation of a queuing system where arrivals follow a Poisson process, service times are exponentially distributed, and there are 'c' servers available to process incoming tasks. This model helps analyze systems in scenarios like customer service, telecommunications, and manufacturing, providing insights into performance metrics such as wait times and system utilization.
Multi-server queue: A multi-server queue is a type of queuing model where multiple servers provide service to incoming entities, such as customers or tasks, simultaneously. This system is designed to improve efficiency and reduce waiting times compared to single-server queues, as it allows for parallel processing of requests. The performance of a multi-server queue is influenced by factors like arrival rates, service rates, and the number of servers available.
Poisson distribution: The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, provided that these events occur with a known constant mean rate and independently of the time since the last event. This distribution connects to several concepts, including randomness and discrete random variables, which can help quantify uncertainties in various applications, such as queuing systems and random signals.
Queue length: Queue length refers to the number of entities waiting in line to receive service from a server in a queuing system. This measurement is essential for understanding system performance, as it impacts wait times, service efficiency, and overall customer satisfaction. The concept of queue length plays a vital role in the analysis of both single-server and multi-server queues, affecting decisions regarding resource allocation and system design.
Reneging: Reneging refers to the phenomenon where customers abandon their wait in a queue before receiving service. This behavior is particularly relevant in scenarios involving both single-server and multi-server systems, as it can significantly impact the overall efficiency and performance of these queues. Understanding reneging is crucial because it influences customer satisfaction, service times, and the overall effectiveness of resource allocation in queuing systems.
Service Rate: Service rate refers to the speed at which a server can process or serve customers in a queuing system, usually measured in units of customers per time period. This concept is crucial because it directly impacts the efficiency and effectiveness of service delivery in both single-server and multi-server queues, influencing wait times and customer satisfaction. Understanding service rate helps analyze system performance, identify bottlenecks, and optimize resource allocation for improved operational efficiency.
Single-server queue: A single-server queue is a queuing model where there is only one server providing service to incoming customers or jobs. This model helps in understanding how systems with limited resources handle varying levels of demand, focusing on factors like wait times, service times, and customer satisfaction. It is fundamental for analyzing performance measures such as average wait time in the queue and system utilization, which are crucial for designing efficient service systems.
Steady-state: Steady-state refers to a condition in a system where key metrics remain constant over time, despite ongoing processes or events. In queueing systems, this concept is crucial as it indicates that arrival and service rates have reached a balance, allowing for predictable behavior. This stability is important in assessing performance metrics like average wait times and system utilization, making it a cornerstone in understanding single-server and multi-server queues.
Traffic Engineering: Traffic engineering is a branch of civil engineering that focuses on the planning, design, operation, and management of transportation systems to ensure safe and efficient movement of people and goods. It involves analyzing traffic flow, studying queues at intersections or service points, and optimizing the use of roadways and transit facilities. Understanding queue systems is crucial for traffic engineers to minimize congestion and improve overall transportation efficiency.
Transient State: A transient state refers to a condition in a system that is temporary and not stable, where the probabilities of being in various states change over time. This concept is crucial for understanding systems that evolve dynamically, especially where states fluctuate before reaching a stable equilibrium or steady-state. In various applications, the transient state helps analyze how systems respond to changes, providing insight into their short-term behavior before they stabilize.
Waiting Time: Waiting time refers to the duration that a customer or item spends in a queue before being served or processed. This concept is crucial in understanding the efficiency of systems involving single-server and multi-server queues, as it affects customer satisfaction and overall system performance. In these queue systems, waiting time can be influenced by factors such as arrival rates, service rates, and the number of servers available to handle requests.
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