Question

Customers arrive at the drive-through window at McDonalds restaurant at the rate of 4 every 10 minutes. The average service time is 2 minutes. The Poisson distribution is appropriate for the arrival rate and service times are exponentially distributed. If a second/new drive-through window is to be considered, only one line would be formed. Also, the clerk at the new window would work at the same rate as the current one. Help the restaurant manager find the following performance parameters:

Answer 1

Let us think about a queue (waiting line) most of us have seen, the line at Orin’s place café in Paccar hall. Let us consider the simplest case, one line in front of a single cashier. For now, ignore the role of second cashier (occasionally open) and the barista. We will discuss in class the wide applicability (not just cafes) of the insights we draw from this model.

Inputs

• Customers arrive at the café and join the line. The time between any two customer arrivals is variable and uncertain but we can say, for example, on average two customers join the line every minute. The rate at which customers join the line is what we call arrival rate (symbol lambda ). This is the same as workload arrival in chapter 1. In this case, arrival rate lambda is 2 per minute. This also means that the average time between two arrivals is ½ minute; we call this interarrival time.

• The time cashier takes to serve one customer is called service time. This too changes from customer to customer and is, therefore, variable and uncertain. Let us say, on average, service time for a customer is 15 seconds = ¼ minute. This also means that the one cashier can serve customers at the rate of 1/(1/4) = 4 customers per minute; this is what we call service rate (symbol mu ).

• Since there is only one cashier, number of cashiers is 1. We use symbol m for this. In case cafe opens another cashier and if a single line is used to feed both servers, we will say number of servers m is 2. In case there are two different lines in front of two cashiers then we will say that these are two different queues, each with m=1. For now, let us go with our original scenario of single queue with single server.

• To summarize, at least three inputs are needed to define a queue: arrival rate (lambda), service rate (mu) and number of servers (m). Note that we are interested in expressing our inputs in terms of rates (per minute, for example) and not in time (minutes, for example).

• Depending on the situation, we may need other inputs describing the extent of variability in arrivals and service. For this basic model, we assume certain type of variability (Poisson distribution for number of arrivals and Exponential distribution for service times) and not worry about it for now.

Outputs

• First output we can get is the utilization of our resource, the cashier. We use the symbol rho for the utilization. This is equal to the ratio of the rate at which work arrives and the capacity of the station. The idea is exactly the same as utilization computation in chapter 1.We know that the rate at which work arrives is arrival rate lambda . One cashier can service the work at the rate of service rate . If there are more than one servers (that is, if number of servers m is more than 1) then total rate at which work can be served, station capacity, will be m multiplied by. Therefore, utilization rho can be calculated as lambda divided by (m multiplied by). . Make sure arrival rate and service rate are expressed in same unit of time; for example, both should be per minute or per hour. In the case of Orin café, . Our cashier is busy 50% of the times. For our calculations to work, utilization rho must be less than 1.

• Second output is number of customers waiting. We use the symbol Lq for this. Note that this excludes the person who is actually getting serviced by the cashier. We have a simple table to find this value. The table is attached. To look at the table we need two things: first, (arrival rate/service rate, that is lambda/mu ) and second, number of servers m. In Orin café’s case is 2/4=0.5 and m is 1. You will find one row in the table that corresponds to these two values. In this row, read the number in column titled Lq. You will see that the number is 0.5. This means that, on an average, the number of people waiting in line Lq is 0.5.

• Only in the special case when there is only one server, m=1, we can also use a simple formula to compute the number of customers waiting Lq. (The table works for all values of m, including m=1). For m=1: . For example, in this case =0.5, same as that we got from table.

• Third output is the time an average customer waits in line to receive service. We use the symbol Wq for this. We have a simple formula to convert number-of-customers-waiting Lq into time-a-customer-waits Wq. To get Wq, divide Lq by arrival rate lambda (Little’s law) In Orin cafe, the time-a-customer-waits min = 15 seconds.

• Sometimes we want to think about the whole system, that is, not just waiting but both waiting and getting service. We would like to know the time-a-customer-spends-in-the-system (we use symbol Ws for this) including both time for waiting and time for service. Clearly, this is equal to time-a-customer-waits Wq plus service time. In Orin café’s case, Ws is just equal to the sum of waiting time (15 sec.) and service time (15 sec.). Ws=30sec = 0.5 min.

• There is also the question of the number-of-customers-in-system (symbol Ls), including both, customers who are waiting and who are getting service. Another application of Little’s law shows that to get number-of-customers-in-system symbol Ls, multiply time-in-system Ws by arrival rate lambda . In Orin café’s case, .

• Finally, to compute the chance that system is idle, that is, there is no customer in the system, we can read the column titled P0 from the table, just the way we read Lq. For Orin café, P0=0.5, that is 50% chance that cashier is free.

Other Performance Measures

• For single-server case, some other performance measures can be computed as following:
• Probability that there are n customers in system
• Probability that wait is greater than t =
• Probability that time-in-system is greater than t =
• For more than one server, spreadsheets are available to compute these measures.

Determining Capacity

• If we increase capacity (by increasing m or by increasing ), we expect that the cost of providing that capacity will increase. We also expect, however, that the customers will wait less and that the cost of customer waiting will decrease. This suggests that we should look at the total cost = (cost of providing service + cost of customer waiting) in order to make decision about how much capacity to provide.
• For example, if Orin café pays $15 per hour to a cashier then adding one more cashier increases the cost of providing capacity by $15 per hour. But it also reduces the number of customer in system from (see above) to ( from table for m=2 and then repeat the above steps to get ). If we assume that a customer’s time is worth $20 per hour then system saves (1-0.533)*$20 per hour = $9.34. Therefore, in this example, from total system cost perspective, we should not add another server.

Other Extensions

• Without making much fuss about it, we have made two significant assumptions about the pattern of variability in arrivals and service: Poisson distribution for number of arrivals and Exponential distribution for service times. These assumptions mean the following: coefficient of variation =(standard deviation / mean) for interarrival times and
• coefficient of variation =(standard deviation / mean) for service times . But what if based on measurement of real data, they are not 1? We call this the case of general arrivals and service.

• It is easy to compute Lq in this more general case as follows:
• Starting from , other performance measures can be computed in the same way as earlier.

Summary

Arrival rate lambda = 1 / (interarrival time, that is time between two arrivals)

Service rate mu = 1 / service time

Number of servers m

Utilization rho  

Assume arrivals Poisson distribution and service time are exponentially distributed.

Average number in waiting line Lq can be obtained from table (given and m)

In case number of servers m=1, we can also use

Average waiting time from Little’s Law

Average time-in-system (waiting time +service time)

Average number-in-system (waiting+getting served)  

Probability that there is nobody in the system P0 is available in table.

For single-server case, m=1, we have following three formulas:

Probability that there are n customers in system

Probability that wait is greater than t =

Probability that time-in-system is greater than t =

Determination of capacity is a trade-off between cost of service capacity and cost of customer waiting.

Coefficient of variation of interarrival times Ca = (Std.dev./mean of interarrival times)

Coefficient of variation of service times Cs = (Std.dev./mean of service times)

If , modify Lq from above by multiplying it with

Alternate

Characteristics of Queuing

• Arrival process: the mean arrival rate per time unit (hour) (X) versus the mean inter-arrival time

(1/X). If 160 customers arrive for service at a bar in an eight hour day,

o What is the arrival rate X per hour?

= 160 Customers/8 hours = 20 Customers/hour

o What is the inter-arrival time — (hour)? A

hour = 1 1 hour 1

= hour /Customer= 20 Customers/hour

-

20 Customers 20

= 0.05 hour/Customer (0.05 hour) (60 minutes) 3 minutes

=

Customeri k hour Customer

=3 minutes/Customer

Please note: the mean arrival rate X and the inter-arrival time la should initially have the same time units,

an hour, for example.

o What is the arrival rate X per 15 minutes?

A = 20 Customers/Hour 5 Customers

4F if teen minutes/Hour

•

Fifteen minutes

o What is the inter-arrival time —9

1 1 Fifteen minutes

A

=

Customers 5 Customers = 3 minutes/customer

5

Fisteen minutes

• Service process: the mean service rate per time unit (hour) (1.1) versus mean service time WO. If

the bar can serve 240 customers in an eight hour day,

o What is the service rate t per hour?

p = 240 Customers/8 hours = 30 Customers/hour

o What is the mean service time — (hour)?

1 1 1 hour

=

it hour = 1

30 Customers/hour 30 Customers 30

hour /Customer=

(1/30 hour) (60 minutes) 2 minutes

Customer) k. hour Customer

=2 minutes/Customer

What are Operating Characteristics of Queuing Theory?

X = mean arrival rate (mean number of arrivals per time unit)

1/X = mean inter-arrival time for arrivals

1.1 = mean service rate (mean number of services per time unit)

141 = mean service time per customer or job

Lq

= average queue length or number of units in line waiting for service

Wq = average waiting time a unit spent in queue before being served

Lq = AWq

3 The average queue length is the arrival rate multiplies by the average time spent waiting

in the queue.

3 Jobs blocked and refused entry to the system are not counted in X.

L = average number of units in the system (Lq in queue plus being served)

W = average time a unit spent in the system (in queue plus being served)

L =

• The average queue length plus the one being served is the arrival rate multiplies by the

average time spent waiting in the queue plus the time being served.

• Jobs blocked and refused entry to the system are not counted in X.

s = number of parallel or equivalent servers in the system

p (Rho) or U = server utilization factor = the proportion of time the server is busy

Pw = Probability of an arriving unit to wait in the queue before being served

Po = Probability of no unit in the system (empty) (neither in queue nor being served)

Pn = Probability of having n units in the system (in queue plus being served)

I HOPE YOU SATISFIED WITH MY ANSWER PLEASE GIVE ME POSITIVE REVIEW

THANKS