Applied Queueing Theory

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A queuing model is not limited to optimizing staffing. It can identify productivity gaps and variations in the process that leads to lower operational efficiency. Once a model is constructed, a series of sensitivity analyses can be performed, including staff productivity impact, staff productivity variations, impact of fast-track line, impact of providers in triage, and other parameters.

Implementing the model was achieved by a multidisciplinary team, including representatives that ranged from frontline clinicians to hospital leadership. Commercially available simulation models have some advantages, but there is value in constructing a basic queuing model in a spreadsheet based on the three correlations. It will make the operation-specific input and output variables transparent to the project team and the client. It also will improve alignment and planning and foster a superior decision process. In addition, most health care systems today have the necessary technology e.

Queuing theory correlations are tested, proven and published by several others. Figure 4 shows a significant improvement to the overall LOS at the emergency department based on a weeklong rapid-cycle testing. Average LOS reduced from minutes to minutes 20 percent reduction by merely aligning the ED staffing plan. For the patient population 1, , process variation, measured in standard deviation, improved from minutes to minutes 20 percent reduction.


The primary change at this facility was the addition of a new provider shift, 10 a. Overall ED provider hours were held constant by reducing provider staffing levels during off-peak hours.

Rogers : Fluid Models in Queueing Theory and Wiener-Hopf Factorization of Markov Chains

Figure 5 shows average LOS by hour, as well as the direct impact of the new provider shift. Hourly LOS improved noticeably from the start of the 10 a. Figure 6 shows a significant improvement to LWBS at another network emergency department. This particular ED was faced with front-end challenges that resulted in long door-to-provider durations.

Consequently, it historically reported higher LWBS data as a result of dissatisfied patients waiting too long to see a provider. Introducing a provider-in-triage process flow at the front end enabled segmentation of patient flow, in which lower-acuity Emergency Severity Index 4 and 5 patients would go through the provider-in-triage, and higher-acuity patients ESI 1,2,3 would continue to follow the main ED path. This segmentation improved overall LOS for lower-acuity patients by 35 percent. The door-to-room and door-to-provider process metrics also improved by 36 percent and 43 percent, respectively.

Such dramatic improvement also was attributed to being able to keep lower-acuity patients vertical; most of them were processed directly from triage to discharge without a bed assignment. LWBS for this facility improved by 58 percent. Standard deviation of daily LWBS percentage for the patient population improved from 6. Queuing theory is an integral part of Lean Six Sigma improvement methodology at Lehigh Valley Health Network, a critical tool during the planning phase that provides a sound framework for process optimization that includes staffing plan productivity.

Given the recent success with its application at network ED locations, Lehigh Valley is planning to replicate similar improvements broadly across the network. Work is already underway at the inpatient unit to unveil defects and variations associated with admission-to-discharge processes. Queuing models also can optimize discrete processes, such as physician practice office flow. Each practice can be modeled and optimized, resulting in less-crowded waiting rooms and faster appointments.

A final note: Queuing model is only a theoretical model that statistically can predict the future operational state based on implementing certain physical, behavioral and work process changes. The model provides only direction and confidence to the improvement team; the team members still must do the heavy lifting of implementation and associated change management. Application of queuing model in isolation, without a strong implementation team, does not guarantee project success. Nevertheless, queuing theory promotes team alignment, team focus and will continue to drive us toward our patient experience goal: approaching zero wait time for our patients.

Naser M. This work would not be possible without the support of leadership, including William Reppy president , John Fletcher president and David Burmeister chair, emergency medicine.

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Once the raw event has been processed, enriched and transformed to an enterprise standard event, the event needs to be published to a Topic provided by the messaging infrastructure. There are many options available for building the enterprise-wide messaging infrastructure a. Kafka is a highly scalable and distributed messaging system gaining lot of popularity among enterprises for Message Oriented Middleware MOM infrastructure. It is important to decouple the event generation and event processing from the customer-facing business processes to avoid any delay in the response time due to the additional processing overhead of event handling.

By putting the raw events in a message queue before the events are processed and published helps in decoupling the business process generating the events and the actual processing of the events, however, this adds the complexity to the overall architecture. There could be a delay in the event processing and publishing if the rate at which the raw events are generated is higher than the rate at which the events are processed and published by event processors.

There could be a business requirement to process the events within X seconds and any delay could result in sub-optimal customer experience or even an undesired consequences. As an example, sending the debit card transaction alert to the customer after 5 mins of swiping the card might not be a good customer experience and a delay of 15 mins in alerting the customer can result in a potential fraud costing a significant expense to the bank.

In order to avoid any delays in event processing, it is important to run performance tests to evaluate the system performance under peak load.

We can also evaluate the system performance before actually running the performance tests by applying the Queuing theory. This helps in setting up a theoretical benchmark against which we can run the performance tests to evaluate the system performance involving messaging infrastructure. Queuing theory has been widely used in Operations Research to calculate the waiting times and the resources required to service customers in call centers, service patients in hospitals and traffic engineering. It is also used in computer science for analyzing the stacks a queue storing system state used for running the processes and resources on the CPU.

An Introduction to Queuing Theory

Any system that involves a queue inherently introduces a delay in serving the customer e. How does your bank know how many agents to staff the call center to service customer calls or how many representatives are required at a bank branch to serve the customers stopping by at the branch? Queuing theory helps in answering all these questions i. Queuing Theory principles can also be applied to a system generating and processing the events. The events are messages waiting in a queue to be served processed by the event processor application hosted on the server. Following are some of the performance metrics that can be evaluated-.

The event arrival follows Poisson Distribution and has an important characteristic of the system being memory-less i.

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The total number of events occurring in a small interval of time is unknown and assumed to be a random variable. The inter-arrival time between the events is also considered to be a random variable i. The inter-arrival wait time for the events follows exponential distribution i. In other words, once the event has occurred the probability that the next event will occur in 1 second is higher than the probability that the next event will occur in ms. The time it takes to process the event is also considered to have an exponential distribution.

The number of servers or the processing nodes that are available to process the generated events.

Jockeying in Queuing Theory

When modeling the event processor application hosted on an application server there could be multiple container threads running in parallel to process the events. Total number of servers i. Markov or memory-less.