Queue length

In reality, the queue size n and waiting time (w) do not behave as a zero-infinity step function at p = 1. Also at lower utilization factors (p < 1) queues are formed. This queuing is caused by the fact that when analysis times and arrival times are distributed around a mean value, incidently a new sample may arrive before the previous analysis is finished. Moreover, the queue length behaves as a time series which fluctuates about a mean value with a certain standard deviation. For instance, the average lengths of the queues formed in a particular laboratory for spectroscopic analysis by IR, H NMR, MS and C NMR are respectively 12, 39, 14 and 17 samples and the sample queues are Gaussian distributed (see Fig. 42.3). This is caused by the fluctuations in both the arrivals of the samples and the analysis times. 

Fig. 42.3. Time series of the observed queue lengths (n) in a department for structural analysis, with their corresponding histograms fitted with a Gaussian distribution.

The following relationships fully describe an M/M/1 system the average queue length (n ) which is the number of samples in queue, excluding the one which is being analysed ... 

The observed Gaussian distribution of the queue lengths in the spectroscopic laboratory (see Fig. 42.3) is not in agreement with the theoretical distribution given by eq. (42.6). This indicates that an analysis station cannot be modelled by the simple M/M/1 model. We will return to this point later. 

In addition to these criteria, resource queue length and resource waiting time are also used in designing some resources. 

Inventories and queue lengths It is often important to know where inventories and queues are distributed through the system. The average total flow time and the average total queue length are connected by Little s law ... 

The modeling of open queueing networks with general service times, is discussed in Chapter 83, Section 7.2. The basis of the approximation is to assume that the queue length distributions at the different service centers are independent, that is,... 

Then again it is possible to show that the queue length distributions are product form and is thus possible to determine the performance measures. If there is just one machine at each service center, then the mean value analysis algorithm can be adapted to find the performance measures. Define n = (rii, Hj,. . . , rip) and n - e, = (nj, nj,. . . , - 1,. . . , rip). Then the expected time that a... 

Since we assume that the queue lengths at the two queues are product form, we have that... 

Measurement of traffic queue length at an intersection high 1,2,3. .. n oars - clear precise precise... 

X = —q, if Ci occupies the q-th place in the queue of components awaiting repair X = 0, if Ci is imder repair X = 1, if Ci is operable and connected to eo X = 2, if Ci is operable and disconnected from eo the sojourn time of ei in the state X coimted from Tk, on the assumption that all components above ei do not change their states before ei does, q len the mnnber of components awaiting repair (queue length),... 

With the number of traffic sources that exist from the start to end of the simulation increasing from 20 to 80, the total probability of cell loss or area under the error signal remains the same. However, the variance of the queue length is considerably reduced with the proposed scheme. The reduction in the variance is more pronounced with large traffic. 

The heuristic rule relative to equipment choice which has the lowest queue length is always chosen (h2 = 4).The batch which enters first the workshop is favoured for h. The best solution 1 obtained at the lib generation is detailed in Table 4. Table 5 presents the comparison between the results obtained by optimization and those obtained by a trial-and-error procedure. 

Monte Carlo technique It is a simulation process. It uses random numbers as an approach to model the waiting times and queue lengths and also to examine the overall uncertainty in projects. 

The loop s execution time obviously depends on the messages in the queue. Thus, a timing analysis needs to know the maximum queue length. It also may have to consider that other tasks may preempt the execution of the loop and add messages to the queue. 

We want to avoid that the queue length for some of the types becomes infinitely long. Therefore it is necessary that for every type the available capacity pa cycle is large enough to produce the average demand during one cycle. This can be described by the following restriction for the capacity set... 

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