# 13.2.2 Logistic Queues as an Effect of Random Load Fluctuations

### Intended learning outcomes: Describe job shop production as a network with work centers as nodes. Explain the average wait time as a function of capacity utilization. Produce a summary of relevant formulas in queuing theory.

With the exception of continuous production, there is no production type in which the capacities of machines and workstations following one another in the process are completely synchronized. As Figure 13.2.1.1 shows, even in other cases of line production, synchroniza­tion is not always possible. Thus, to a certain extent, buffers serve to balance the differing output rates of the work centers and to ensure continual load of the individual work centers over a certain period of time.

Fig. 13.2.2.1       Job shop production as a network with work centers as nodes.

These buffers are queues formed in front of a workstation; the size of the queues changes over time. Particularly in job shop production, there is great variation in the behavior of the buffer, since a queue is fed from many locations. We can view job shop production as a network with work centers as nodes, as represented in Figure 13.2.2.1. In the figure, the nodes represent work centers, which are classified as homogene­ous. The arrows represent the flow of goods or information be­tween these work centers. In the discussion below, the focus is on “Node I” of this network.

Input enters from various nodes and sometimes also from the outside (from a store or a receiving department, for example). This input arrives at a joint queue in front of one of the various workstations (S1, S2,  . . . ,  Si) of work center i. After completion of the operation in Node i, the orders flow to other nodes or toward the outside, either in part or in their entirety (after a final operation), depending on the specification in the routing sheet. In line production, there is essentially a sequence of nodes rather than a network.

As mentioned above, determining the size of a buffer is an optimization problem. Queuing theory provides some fundamental insights into the way that job shop production functions and, to a certain extent, how line production functions as well. Here we limit our discussion to the stationary state of a queue, that is, the state after an infinite time period and with fixed constraints.

For the following discussion, Figure 13.2.2.2 sets out several definitions of variables from queuing theory.

Fig. 13.2.2.2       Definitions of queuing theory variables.

To simplify the discussion, assume the following:

• Arrivals are random; that is, they follow a Poisson distribution with the parameter λ. λ is the average number of arrivals per period under observation.
• Arrivals and the operation process are independent of one another.
• Execution proceeds either in order of arrival or according to random selection from the queue.
• The duration of the operations is independent of the order of processing and is subject to a determinate distribution with mean M(OT) and coefficient of variation CV(OT).

Figure 13.2.2.3 shows the average wait time as a function of capacity utilization for a model with one station (s = 1, where a queue feeds only one operation station, i.e., one workstation or one machine). We assume the coefficient of variation CV(OT) for the distribution to be 1, which is the case with a negative exponential distribution, for example.

Fig. 13.2.2.3       Average wait time as a function of capacity utilization: special case s = 1, CV(OT) = 1.

Exercise: Get used to the effect of queues by choosing different values for the queuing theory variables.

Figure 13.2.2.4 presents the relevant formulas of queuing theory for the average case, with references to their original sources in the literature, specifically [GrHa18], [Coop90], and [LyMi94], including page and formula numbers. For further aspects of theoretical mathematics, the reader can consult [Fers64] and [Alba77]. It is important to note, however, that for multiple-station models (s = arbitrary), the relationships based on numerical calculation only approach validity under conditions of extensive capacity utilization.

Fig. 13.2.2.4       Summary of relevant formulas in queuing theory.

Figure 13.2.2.5 shows wait time as a function of operation time for selected values of s and CV(OT).

Fig. 13.2.2.5       Average relative wait time as a function of capacity utilization: selected values (following an unpublished slide of Prof. Büchel, ETH Zurich).

## Course section 13.2: Subsections and their intended learning outcomes

• ##### 13.2 Logistic Buffers and Logistic Queues

Intended learning outcomes: Explain wait time, buffers, the funnel model, and queues as an effect of random load fluctuations. Present conclusions for job shop production. Produce an overview on logistic operating curves.

• ##### 13.2.1 Wait Time, Logistic Buffers, and the Funnel Model

Intended learning outcomes: Describe inventory buffers to cushion disturbances in the production flow. Explain the buffer model, the reservoir model and the funnel model.

• ##### 13.2.2 Logistic Queues as an Effect of Random Load Fluctuations

Intended learning outcomes: Describe job shop production as a network with work centers as nodes. Explain the average wait time as a function of capacity utilization. Produce a summary of relevant formulas in queuing theory.

• ##### 13.2.3 Conclusions for Job Shop Production

Intended learning outcomes: Present qualitative findings of queuing theory for job shop production and, in part, for line production. Describe the measures indicated by the qualitative findings of queuing theory.

• ##### 13.2.4 LOC — Logistic Operating Curves

Intended learning outcomes: Produce an overview on logistic operating curves. Explain an example of logistic operating curves.