Suppose that the cumulative number of demands for service or "customers" at any time t is known and equal to the value of the function A(t). These "customers" might be crane loads, weld inspections, or any other defined group of items to be serviced. Suppose further that a single server is available to handle these demands, such as a single crane or a single inspector. For this model of queueing, we assume that the server can handle customers at some constant, maximum rate denoted as x "customers" per unit of time. This is a maximum rate since the server may be idle for periods of time if no customers are waiting. This system is deterministic in the sense that both the arrival function and the service process are assumed to have no random or unknown component.
Figure 4-6: Cumulative Arrivals and Departures in a Deterministic Queue
A cumulative arrival function of customers, A(t), is shown in Figure 4-6 in which the vertical axis represents the cumulative number of customers, while the horizontal axis represents the passage of time. The arrival of individual customers to the queue would actually represent a unit step in the arrival function A(t), but these small steps are approximated by a continuous curve in the figure. The rate of arrivals for a unit time interval t from t-1 to t is given by:
4.14
While an hour or a minute is a natural choice as a unit time interval, other time periods may also be used as long as the passage of time is expressed as multiples of such time periods. For instance, if half an hour is used as unit time interval for a process involving ten hours, then the arrivals should be represented by 20 steps of half hour each. Hence, the unit time interval between t-1 and t is t = t - (t-1) = 1, and the slope of the cumulative arrival function in the interval is given by:
4.15
The cumulative number of customers served over time is represented by the cumulative departure function D(t).
While the maximum service rate is x per unit time, the actual service rate for a unit time interval t from t-1 to t is:
4.16
The slope of the cumulative departure function is:
4.17
Any time that the rate of arrivals to the queue exceeds the maximum service rate, then a queue begins to form and the cumulative departures will occur at the maximum service rate. The cumulative departures from the queue will proceed at the maximum service rate of x "customers" per unit of time, so that the slope of D(t) is x during this period. The cumulative departure function D(t) can be readily constructed graphically by running a ruler with a slope of x along the cumulative arrival function A(t). As soon as the function A(t) climbs above the ruler, a queue begins to form. The maximum service rate will continue until the queue disappears, which is represented by the convergence of the cumulative arrival and departure functions A(t) and D(t).
With the cumulative arrivals and cumulative departure functions represented graphically, a variety of service
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indicators can be readily obtained as shown in Figure 4-6. Let A'(t) and D'(t) denote the derivatives of A(t) and D(t) with respect to t, respectively. For 0 t ti in which A'(t) x, there is no queue. At t = ti, when A'(t) >
D'(t), a queue is formed. Then D'(t) = x in the interval ti t tk. As A'(t) continues to increase with increasing t, the queue becomes longer since the service rate D'(t) = x cannot catch up with the arrivals. However, when again A'(t) D'(t) as t increases, the queue becomes shorter until it reaches 0 at t = tk. At any given time t, the queue length is
4.18
For example, suppose a queue begins to form at time ti and is dispersed by time tk. The maximum number of customers waiting or queue length is represented by the maximum difference between the cumulative arrival and cumulative departure functions between ti and tk, i.e. the maximum value of Q(t). The total waiting time for service is indicated by the total area between the cumulative arrival and cumulative departure functions.
Generally, the arrival rates At = 1, 2, . . ., n periods of a process as well as the maximum service rate x are known. Then the cumulative arrival function and the cumulative departure function can be constructed systematically together with other pertinent quantities as follows:
1. Starting with the initial conditions D(t-1)=0 and Q(t-1)=0 at t=1, find the actual service rate at t=1:
4.19
2. Starting with A(t-1)=0 at t=1, find the cumulative arrival function for t=2,3,. . .,n accordingly:
4.20
3. Compute the queue length for t=1,2, . . .,n.
4.21
4. Compute Dt for t=2,3,. . .,n after Q(t-1) is found first for each t:
4.22
5. If A'(t) > x, find the cumulative departure function in the time period between ti where a queue is formed and tk where the queue dissipates:
4.23
6. Compute the waiting time w for the arrivals which are waiting for service in interval t:
4.24
7. Compute the total waiting time W over the time period between ti and tk.
4.25
8. Compute the average waiting time w for arrivals which are waiting for service in the process.
4.26
This simple, deterministic model has a number of implications for operations planning. First, an increase in the maximum service rate will result in reductions in waiting time and the maximum queue length. Such increases
might be obtained by speeding up the service rate such as introducing shorter inspection procedures or installing faster cranes on a site. Second, altering the pattern of cumulative arrivals can result in changes in total waiting time and in the maximum queue length. In particular, if the maximum arrival rate never exceeds the maximum service rate, no queue will form, or if the arrival rate always exceeds the maximum service rate, the bottleneck cannot be dispersed. Both cases are shown in Figure 4-7.
Figure 4-7: Cases of No Queue and Permanent Bottleneck
A practical means to alter the arrival function and obtain these benefits is to inaugurate a reservation system for customers. Even without drawing a graph such as Figure 4-6, good operations planners should consider the effects of different operation or service rates on the flow of work. Clearly, service rates less than the expected arrival rate of work will result in resource bottlenecks on a job.