1. Consider the single-server queue with geometrically distributed inter-arrival
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Question
1. Consider the single-server queue with geometrically distributed inter-arrival times and service
times as discussed in class, with a departure rate of = 0:6. Plot the expected delay of a job
as a function of the arrival rate . You have to run the simulations long enough, for example
at least for 106 time slots.
(Hints: use Little's law to calculate the expected delay, but you have to simulate the queue
dynamics to get the expected queue length rst.)
Note:
Choose one programming language to do the simulation from C/C++, Matlab, Python,
and Java.
Report should include the plot, and simulation code and annotations.
Explanation / Answer
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 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 ).
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.
To understand the effects of nonmandatory retirement age on new faculty hiring, we apply Little’s Law of Queueing from operations research:
L = W, (1)
where
L = the time-average number of “customers” in the queueing system,
= average rate at which new customers enter the system, and
W = the average time spent in the system by a random customer who enters.
Little’s Law applies to any queueing system operating in steady state (Little 1961). The relationship does not require Markovian or other simplifying structural assumptions and is valid independent of probability distributions involved. It has stood the test of time, as discussed in a recent historical overview article celebrating its 50th anniversary (Little 2011). In application to university faculties, we have the following correspondences:
L = the time-average number of tenure-track faculty members employed by the university,
= average annual rate at which new tenure-track faculty members join the faculty, and
W = the average number of years spent on the faculty by a newly hired assistant professor.
For this initial analysis, we focus only on tenure-track appointments and ignore more senior faculty appointments above rank of assistant professor. We believe that ours is the first application of Little’s Law to college and university faculties.
To apply Equation (1), we need to estimate W, the average number of years spent on the faculty by a typical new assistant professor. To do this, we think of the key decision points made for and by this faculty member as she or he proceeds through her or his career. A typical career trajectory involves the following milestones:
At the end of year 2, reappointment as assistant professor without tenure.
At the end of year 4, appointment as associate professor without tenure.
At the end of year 7, appointment as associate professor with tenure.
Remaining on the faculty until retirement, not voluntarily departing from the university at some preretirement time, most often associated with a faculty or business appointment elsewhere.
Retirement from the faculty.
Milestone 4 is not really a specific time-marked event. It is included to account for those faculty, typically in their 40s or 50s, who voluntarily leave to become deans, provosts, and presidents at other universities or to engage full time in some business activity. At MIT, an example is Dr. Robert Brown, once provost of MIT, who left in 2005 at age 54 to become president of Boston University. Another example is Dr. Lawrence Bacow, once chancellor at MIT, who in 2001 at age 49 became president of Tufts University.
Let us denote
Pi = conditional probability that milestone i is achieved, given that milestone(i 1) is achieved, i = 1, 2, 3, 4, 5.
Then, of course, (1 Pi) is the conditional probability that milestone i is not achieved, given that milestone (i 1) is achieved. We assume that if any reappointment, promotion, or tenure decision (milestone 1, 2, or 3, respectively) is unsuccessful, institutions allow the affected faculty member to remain on the faculty for one more year after that negative decision to allow time for that individual to identify and solidify alternative career paths. We assume that the new faculty slot created by such a negative decision opens up only after the individual has left the institution’s faculty. For the sake of simplicity, we will also assume that P5 = 1, meaning that any tenured professor who does not voluntarily leave the faculty eventually retires with probability 1.
For the times associated with milestones, we define for i = 1, 2, 3, 4, 5,
Yi time from first appointment as assistant professor until milestone i is achieved.
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