#### Executive Summary

Queuing theory is an important theory in Operations Research that aims to analyse the performance of waiting line systems. It uses principles of statistics, probability, and algebra to study important characteristics of the system like average queue length, average system length.

In this project, I have tried to apply queuing theory to a socially relevant situation in a COVID vaccination center in the city of Mumbai in India. This center was vaccinating senior citizens who had to wait in queues before getting their vaccine.

The study measures the arrival rate and the service rate of citizens at the vaccination center. Then the queuing theory formulae have been applied to find the queue length and the system length which are socially sensitive in this case. A longer queue and system length for senior citizens would mean that they have to wait longer in the system and are more likely to get in contact with others which would increase the possibility of contracting the virus. It would also be strenuous for them to stand in the queue as they would not always get a place to sit. My motivation was to see how successful the theoretical queuing model is at predicting the actual values.

Another important aspect that I explored was the proof of the queuing theory formulae. I derived every formula used in the essay from basic principles. It was a really challenging task given the multidisciplinary nature of the mathematics involved. For example, there were concepts from statistics involving study if Poisson and Exponential distribution. There was derivation of average values from probability distributions which involved sums of infinite series. At the end of the project, I felt enriched at having studied and explained these cryptic concepts in a lucid way.

I measured the system characteristics like the system utilization, queue length, and system length from actual values. Then I applied statistical measures to test whether the observed values were matching with the values predicted by the theory. Using t-test, it was noted that the observed values matched with the theoretical values within 5% level of significance. It proved that the queuing theory is very much applicable in this scenario.

In future, I would like to explore this topic more by trying to see how the system can be improved to increase the system utilization and hence the efficiency. I would also like to explore reducing the queue and the system length so that it is more convenient for the citizens to get a smooth vaccination experience.

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