Starbucks Queuing Model
Inevitably, for those coffee drinkers that don’t have a coffee machine at home, it is a fact of life that you would have to wait in lines at your closest coffee shop to get that glorious cup of Joe. For the one of the most prolific coffee chains, Starbucks, this is a problem that needs to be addressed as customer service is one of the cornerstones of the industry. In order maintain a good customer service, Starbucks must minimize the in-line wait time and the service times so that customers can go in and out of the shop at their own leisure. Faster service means more customers can be helped in a short amount of time as well as keeping lines short during rush-hour times. Shorter lines, in turn, means that as customers enters the store they will be less likely to be discouraged by the amount of waiting they would have to do and leave for other source. The best way that this problem may be resolved is through analyzing and simulating the queuing behavior of customers as they enter store to make their purchases. Through observations and statistical data, we seek to utilize a model to simulate this behavior as well as determine what variables are critical to minimizing customer in line waiting times as well as customer service times.
Background
Starbucks was formed in Seattle, Washington back in 1971 by three partners. They originally were not a coffee shop. Rather, they were a provider of whole coffee beans to restaurants and other coffee bars. It was not until 1987 that Starbucks transformed into the everyday coffee shop that we know today. Over the course of almost 30 years, Starbucks has expanded to over 60 countries and opening just over 21,000 individual stores. In Q2 of the 2015 fiscal year, they had a total consolidated revenue of $4.6 billion dollars while opening an additional 210 stores worldwide. Their product mix includes high quality roasted coffees, tea, fresh food items and other beverages. They also sell a variety of coffee- and tea-related products such as cups and license their trademarks through other channels such as licensed stores, grocery and national foodservice accounts. Starbucks also markets its products mix with other brand names within its portfolio of companies, which include Teavana, Tazo, Seattle’s Best Coffee, Starbucks VIA, Starbucks Refreshers, Evolution Fresh, La Boulange and Verismo.
Data Collection
Our team members observed 60 customers over the course of two hours, on Friday morning, 9am-11am at the Starbucks shop on the corner of Blaine and Iowa, Riverside. We set up a laptop and used a simple stopwatch application to record customer arrival and wait times in Excel. After gathering this data, we used it as well as Poisson distribution and M/M/K theory to analyze the situation.
Model selection
In our observation at Starbucks, we figure out people arrive according to a certain distribution. The service provided by barista is also provided according to certain distribution. Moreover, at the Starbucks we surveyed at, its queuing system appears to be a single server model with finite queue length. As a single lane system, customers get into a common line and as soon as the front customer leave, the next customer will join get his or her coffee. This common line has limitation on the length of the queue, so sometimes customers will leave if the waiting is longer than their expectation (we will discuss balking in final paper). In our observation, we collected inter-arrival period and service period of 60 customers to represent the infinite population of Starbucks customers. We also observed that arrivals follow a Poisson distribution, with arrival rate /h, and service rate follows exponential distribution at /hr . Meanwhile, Poisson distribution and exponential distribution are related. In our case, we observed arrival rate by service rate is 2.41, which is greater than 1. Therefore, the queue length will automatically increase every minutes due to some customers who join the queue will never get served within that time frame.
There are four important parameters analyzed in our case: length of the system, length of the queue, waiting time in system, and waiting time in queue. Length of the system () is the expected number of people who are actually in the system, including the person who is being served. Length of the queue () is the expected number of people who are waiting for service in this system. Customers are also interested in waiting time for them to get coffee and leave Starbucks () and there are times customers are interested in waiting time to get coffee(). All four of these parameters are actually related to each other, and queuing system finally is going to be measured for equations that define each of them. These are expected values, so they are depend on the probability that amount of people in this system. Our task is derive expressions for the four expected values from probabilities of different amount of customer present in this system.
Poisson Distribution
Any time there is more customer demand for a service than can be provided, a waiting line occurs. Customers can be either humans or inanimate objects. In a waiting line system, managers must decide what level of service to offer. A low level of service may be inexpensive, at least in the short run, but may incur high costs of customer dissatisfaction, such as lost future business and actual processing costs of complaints. A high level of service will cost more to provide and will result in lower dissatisfaction costs.
Because of this trade-off, management must consider what the optimal level of service to provide is. The challenge is to design a service system with adequate but not excessive amounts of capacity. A manager cannot be sure how much customer demand there will be, and it does not know exactly what each customer will order—each order can be unique and require different service time.
Poisson distribution can be used in the case of very large samples. Your hypothesis was that you would find 'x' (the top cell) occurrences of a phenomenon, whereas in fact you found 'n' (the second input cell). For example, the phenomenon might be the number of cases of a rare disease, the number of accidents on a busy junction, the number of stoppages on a production line, or the number of ethnic children at a football match. You want to test the assumption that environmental and other factors influencing the phenomenon were constant between observation periods. The program echoes the point probability and the probability of there being 'n' or more occurrences of a phenomenon given your expectation of 'x' occurrences. For the Poisson distribution you do not need to give a sample size. If the sample size is known, it is generally preferable to use the Binomial. The Poisson distribution is the canonical distribution of a Poisson process where there the frequency of events is independent of the events that have already happened, e.g. radioactive decay, etc. Its safe bet that the number of chips in a cookie is independent of how many chips have already made it into each dollop of delicious cookie batter. In theory, Poisson-distributed arrivals typically experience wait times.
A Poisson process in any process that follows a Poisson distribution. The kinds of processes include the number of customers arriving per minute at Starbucks. That is, it is a count of random events that are not too many. We are looking at a handful of arrivals per minute then the Poisson distribution is a good model.
We simulate the number of customers arriving at Starbucks using a Poisson distribution. We assume that the average over our experiment is 3.7 customers per minute. We would expect to count 3 arrivals per minute about 21% of the time. We would expect to count 0 and 8 arrivals per minute about 2% of the time.
The M/M/1 model and Exponential distribution
M/M/1 is the most basic model among all queuing models. To be more specific, the first M is arrival rate, the second M is service rate, and 1 means 1 server. We use Poisson distribution to calculate arrival rate. As mentioned prior, we are looking at only a handful of arrivals per minute therefore the Poisson distribution is a good model. Moreover, we use this instead of normal distribution because people use Poisson distribution when arrival rate satisfies conditions of orderliness, stationarity, and independence. More precisely, orderliness means that in a time instant, only one customer can arrive. Stationarity means that in a time frame, the possibility of a customer comes within a time interval is as same as time intervals for the same length. Independence means that no matter how many customers come within a time interval, the number of customers in other time intervals will not be affected.
Since the time of brewing a cup of coffee is about the same, we use exponential distribution to calculate it. Actually, in most other business situation, service times are widely modeled by random variable. Due to the relative homogeneity of the brewing process, we can safely assume that the time to brew and serve a cup of coffee to a guest will be equal.
M/M/K queuing system
M/M/K queuing system is the system which has infinite customer population and infinite waiting line. In other words, there will be continuous customers who coming in and waiting in the line. The main difference between the M/M/1 model and this is the fact that the M/M/K model is used in cases where there are multiple servers. The K in the model name represents this fact. The best examples will be free-way toll road, and subway stations, which are public facilities providing transportation services 24/7. Followings are conditions should be met:
1. Customers arrive according to a Poisson process at mean rate
2. Service times follow an exponential distribution
3. Each of the k servers works at an average rate m
Poisson Arrival Distribution (Probability of k Arrivals within time t):
P(X=k)=

l= the mean arrival rate per time unit t= the length of the time interval e= 2.7172818 (the base of the natural logarithm) k!= k(k-1)(k-2)(k-3)…(3)(2)(1)
In this expression, l and t must be expressed in the same time units; that is, if l is expressed in customers per hour, t must be expressed in hours.
Exponential Service Time Distribution f(t)= μ

m= the average number of customers who can be served per time period
Balking
A single waiting line can maintain fairness for customers waiting in line. After all, a person who entered the store before another should expect to be served first. However, it does have it disadvantages when compared to a multi-lane line. For example, if a bank has only three tellers working on a Friday afternoon and 30 customers are present in the bank, 27 people are waiting in a single waiting line. Customers perceiving the waiting line as too long may balk (not join the queue) and decide to take their business elsewhere. If too many customers balk, the long term profitability of the business will suffer. On the other hand, if there were an individual line for each teller, the average number of customers waiting in each line for each teller would be nine. While a line length of nine is certainly long, it is not as intimidating as 27, and customers may be less likely to balk. Therefore, we need to balance fairness with efficiency.
M/M/K/F model
In cases such as Starbucks, customers may balk because of waiting line is longer than they expected. In such cases, we will use M/M/K/F, which formula uses infinite customer population and finite waiting line. Thus, F represents the maximum number that can wait in line. Should maximum capacity be achieve, any new customers will be forced to leave. The rejected customer will then leave the system and is considered irrelevant to the model.
Effective Arrival Rate for Finite Queues
= λ(1-PF)

Excel Simulation
Input data Arrival rate (lamda) | 0.01150 | Service rate (mu) | 0.00477 | Number of servers | 3.00000 | Mean interarrival time | 86.94915 | Mean service time | 209.57627 | Standard deviation of interarrival times | 73.83029 | Standard deviation of service times | 103.00790 | Coefficient of Variation arrive | 0.72101 | Coefficient of Variation service | 0.24158 | u | 2.41033 | Server Utilisation (Ro) | 0.80344 | Average time waiting in line (Wq) | 111.66691 | Average number of customer waiting in line (Lq) | 1.28428 | Average time in system (W) | 321.24318 | Average number of customer in system (L) | 3.69461 |

Arrival rate equals to one over average inter-arrival time. Service rate μ equals to one over average servers time. In our model, we want to compare range of average time in system and average time of customer in system to determine the optimal servers number. We first use data table to run number of servers against service time sigma to find out average time in system. Based on the data table, the largest marginal change take place when servers number is increased to 4. EMBED MSGraph.Chart.8 \s Moreover, we assume each Starbucks server receives minimum wages of $9 per hour. Starbucks value each customer’s waiting cost for $20 per hour. The total cost equals to server cost plus customer waiting cost, and to minimize the cost Starbucks should have 4 servers for the morning period. EMBED MSGraph.Chart.8 \s Lastly, we assume buffer size of the Starbucks on Blaine and Iowa will be 10 customers. Based on this buffer size, we assume more than 4 servers could make sure the balking rate stays below 1%. Buffer (waiting space) size | 10 | | | | | | | | | | | Servers | 2 | 3 | 4 | 5 | 6 | Probability of no customer in the system (P0) | 0.0930 | 0.0550 | 0.0820 | 0.0880 | 0.0894 | Probability of rejecting a customer (balking) | 100.00% | 1.44% | 0.07% | 0.00% | 0.00% |

From the data collected from our Excel equations, we can conclude that in the situation where balking occurs the number of servers once again follows the findings we see previously. At two servers, despite the higher probability of having no customers in the system, we see that the probability of being forced to lose patronage is 100%. In other words, this means that two servers will be insufficient in handling their orders in a timely fashion. Customers will enter store only to be forced to leave as there will be no room left for them. As soon as a third server is added to the shop, the probability of rejecting a customer dramatically falls to 1.44%. The difference between two servers and three servers once again serves as the threshold into efficiency. Increasing the number of servers at this point only serves to increase costs and return for negligible gains in decreasing the probability of customer rejection.

Conclusion

Based upon our findings, we can arrive to the final conclusion that the optimal number of servers is four servers. We see that we can gain the greatest marginal savings as well as the lowest cost of service with the four. Given that customer behavior with balking means that there is the greatest marginal benefit at three servers, nevertheless the four server balking rate in conjunction the cost calculation means that four hire’s is the overall best decision for Starbucks. One thing that must be taken into account is that the data that our conclusion is derived from is only one period of time for one specific store. In order to gain a truly comprehensive recommendation, more data points must be gathered and analyzed at a variety of stores as well as different time periods. Due to this we cannot speak for every single Starbucks store out there, we can only speak for store where we gathered our data. Our final recommendation for the Starbucks on Blaine and Iowa is to hire four servers so that they may provide the best service for the minimal costs.