Queuing and Waiting

No queues would be formed if no new samples are submitted during the time that the analyst is busy with the analysis of the previous sample. If all analysis times were equally long and if each new sample arrived exactly after the previous analysis is finished, the analytical facility could be utilized up to 100%. On the other hand, if samples always arrive before the analysis of the previous sample is completed, more samples arrive than can be analyzed, causing the queue to grow indefinitely long. Mathematically this means that  

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. 

The number of arrivals follows a Poisson distribution when samples are submitted independently from each other, which is generally valid when the samples are submitted by several customers. The probability of n arrivals in a time interval t 

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. 

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