# Queuing Theory

Queuing theory is a mathematical analysis of waiting times and queues in stochastic systems. In essence, queues will arise when demand in the short-run exceeds capacity or the ability to provide service. Majorly, queuing theory is used to in the analysis of production and service processes that show variability observed to be random in the demand or arrival times and the time taken to provide such a service. If the system is expanded to cover periods, in the long run, the queue will not have the capacity to accommodate such demand, and as such, the model will explode (Ferreira et al. (2012). The queuing model can be applied in systems such as banking, taxi cabs, fire engines, elevators, conveyor belts, inspection stations in an airport, waiting lists in organ transplant, hospital service and in judicial processes among many others.

Managers or system designers often need queuing analysis to help them come up with a system that will effectively balance the cost of increased capacity as compared to the benefits of increased productivity in service delivery. According to Kalashnikov (2013), the systems thus implemented should help system administrators to reduce waiting costs as well as the cost of lost sales attributed to waiting. For instance, in a hospital where patients arrive on their own accord or using ambulances, with a doctor always on duty, a model can be developed to ensure that service delivery is optimized. The model should provide critical answers to the management. Such explanations include whether the management should add another doctor or not, how many deaths can be attributed to the waiting among many other considerations. The cost of service should as much as possible be matched to the costs of waiting as this is the turning point for total costs.

Any queuing system has three main components, the calling population, the arrival process and he queue configuration as explained by Ferreira et al. (2013) . These three components are the variables in the mathematical formulation of any queuing model. The calling population that specific population from which customers come from and be finite or infinite. For instance, in a hospital situation, patients are infinite, you cannot tell when they will be coming to an end. The arrival process determines the when the how and where customers or jobs arrive at the system. An important consideration is the arrival interval, and it is determined through extensive data collection and analysis. The queue configuration determines the number of queues that should be in the system. It often determines the customer behavior, for instance, jockeying and balking. The limits of such a queue should be determined.

Mainly there are single line and multiple line systems. Single line systems ensure that there is fairness to all customers, mostly o first come, first served basis, second, they provide no option for jockeying and customer anxiety which can be fatal in some systems, for instance, in traffic and third, it helps to solve the “cutting in” problem. The multiple line system provides the user with faster service times. Two, services can be differentiated, for instance, a supermarket can have an express lane for clients with less than five items. Third, labor can be specialized, for example, in a general hospital we can have a queue for kid patients or adult patients. Four, balking tendencies can be deterred as well as providing customers with more flexibility.

The M/M/1 Model

The M/M/1 model is the simplest queuing model and has the following assumptions;

• Calling populations are infinite
• Arrival times of the customers are independent
• There is an expected arrival rate λ in the arrival process which is Poisson
• The queue is single with an infinite length and no options for reneging or balking
• The service time is µ and is exponentially distributed
• The discipline in the queue is always maintained at first come, first served basis.

Example,

In a train station, customers are checked in by a single scanning machine. The machine checks in 412 passengers per hour. 320 passengers arrive at the station per hour. Using the M/M/1 model, it can be satisfactorily established that, the system is 77.67% utilized, the queue is expected to have about four people at any given time, customers wait in the queue for half a minute and the expected total time in the system is 40 seconds. The probability that a customer will wait is 0.7767. To properly see the associated graph please refer to the attached excel sheet, tab MM1-Train. From the above example, it can be fairly possible for the management to determine when they will increase the number of check-in machines.

In the real world, the design of queuing systems always involves some capacity decisions; service time, number of service stations and number of servers per station as indicated by McManus et al. (2004). Managers should always have these two questions in mind when designing systems, what service level is most suitable for our business? And what is the capacity to be acquired? With the attached excel sheet, managers can model different scenarios that will provide them with answers to their questions. They should be careful not to have unreasonable idle time due to excess capacity. In short, the cost of increasing capacity should be balanced against cost reduction resulting from shorter waiting time. To achieve this, the management must,

1. Specify the waiting cost for making customers wait. This is often the difficult part and should be reflective of the impact in dollar terms that a delay has on the organization.
2. Specify a level of service that is acceptable and hence reduce the capacity under such a condition.

If the costs above are determined, then the function can be modeled as TC=WC+SC. The objective should be minimizing TC where

• TC is the Total Cost
• WC is the Waiting Cost
• SC is the Service Cost

Waiting Cost should be given as a function of the product of Waiting Cost per Customer and time unit and the number of customers in the queue.  i.e. WC=Cw*Lq. Service Cost is a function of the product of number of servers, c, and the expected cost per server per unit time, Cs, and the average time taken to serve one customer, µ. i.e. SW=c*Cs* µ.

Using our example above, we can compute the total cost associated with the model. Take Waiting Cost to be \$0.05 per customer and SC to be \$0.04. it therefore means that the TC to be \$0.05+\$0.04=\$0.09

In conclusion, the queuing model is an important element that management in any industry can exploit to align their production or service delivery to customer needs. Calculation of important variables is important to ensure that there is credible information to that effect. Data collections and analysis is part of the ground leveling to ensure that proper information that can be manipulated is availed.

References

Ferreira, M. A. M., Andrade, M., Filipe, J. A., & Coelho, M. P. (2012). Statistical queuing theory with some applications. International Journal of Latest Trends in Finance and Economic Sciences, 1(4).

Kalashnikov, V. V. (2013). Mathematical methods in queuing theory (Vol. 271). Springer Science & Business Media.

McManus, M. L., Long, M. C., Cooper, A., & Litvak, E. (2004). Queuing theory accurately models the need for critical care resources. Anesthesiology: The Journal of the American Society of Anesthesiologists, 100(5), 1271-1276.

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