lecture1cumdiag - Cumulative Diagrams: An Example Consider...

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Cumulative Diagrams: An Example Consider Figure 1 in which the functions O (t) and P (t) denote, respectively, the demand rate and the service rate (or “capacity”) over time at the runway system of an airport 1 . The demand rate is the expected number of demands per unit of time, i.e., the number of requests for service at the runway system per unit of time. Note that not all these requests will be necessarily satisfied due to capacity limitations. Similarly the service rate is the expected number of service requests that can be satisfied per unit of time when the runway system is continually busy, i.e., it is the maximum throughput capacity of the runway system (details in the next lecture of our course). The units we shall use for both the demand rate and the capacity are “aircraft movements (arrivals or departures) per hour”, unless otherwise stated. Figure 1 approximates a situation that one sees all the time at busy airports. The time axis shows the busy part of a day at the airport, e.g., the origin may correspond to 07:00 local time. To simplify the analysis we assume the demand rate is constant throughout the busy part of the day, i.e., (t) = n movements per hour for all t . While the normal capacity h (for “high”) is greater than n , a weather event occurs between t = a and t = b (e.g., this could be a period when fog is present during a particular day) which reduces the capacity to l (for “low”) movements per hour. It will be convenient later on to denote with T = b – a the interval of time during which the capacity is low. We would like to explore quantitatively the implications of this temporary loss of capacity on delay levels at the airport. 1 We shall use throughout the Greek symbols O# and to denote the expected “demand rate” and expected “service rate” (or “capacity”); this is consistent with the standard notation used in queuing theory.
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We shall make another important simplification in our model at this point. Namely, we shall assume that demands occur at evenly spaced intervals (e.g., if n = 60, a demand occurs exactly every 60 seconds) and, similarly, for any value of the service rate, h or l , the service times are constant (e.g., if l = 30 and the runway system is continually busy, an aircraft movement is completed exactly every 2 minutes). In queuing theory, this is called a deterministic model, by contrast to probabilistic models in which either the times between the occurrence of successive demands or the service times (or, more usually, both) are random variables, i.e., may take on different values, each value associated with a certain probability. Clearly the situation shown in Figure 1 will result in some delays to aircraft movements during the time period between t = a and t = b , at the very least. In fact, it is very easy to plot, as a function of time, the number of aircraft (landing or taking off) which will be in the queue, waiting to use the runway system. This is done in Figure 2. There is no queue until
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lecture1cumdiag - Cumulative Diagrams: An Example Consider...

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