Systems With Limited Capacity

Systems With Limited Capacity

In the models studied in the earlier chapters, we assumed that there is enough space for the customers to line up before a station; however in reality this assumption is not always the case. In a factory, usually the space for work-in-process inventory in front of a station is limited. In hospital waiting rooms or restaurants there is space for a certain number of customers. In a system such as stadium ticket offices or candy stalls in theaters and movie theaters, the area available for customers to line up is finite, however said area is so ample that it is generally not taken into account in the calculations.

Passageways and corridors in buildings or roads and highways are examples of systems that are represented as queueing systems with finite capacity. In these cases, the number of vehicles that can use the road (server) depends on the characteristics of the section (length), therefore the number of vehicles that can enter the system is restricted to a certain amount.

One indicator that is constantly watched by the administrators is the amount of work in process that accumulates before the stations, and usually maximum limits are set for the pieces lined up. The reasons for keeping or setting limits are a matter of space or economics.

An area where there is a fixed maximum number for waiting customers is normally easier to control than one where customers are allowed to arrive and line up. The work-in-process inventory must always be watched as it represents a monetary investment in raw materials.

When it is necessary to put a limit on the amount of work in process for any reason, this will trigger a series of effects that need to be considered, as they will have an impact on a system’s indicators.

Figure 1.

A production line with buffers

Let us consider a production line like the one in Figure 1. The system consists of n stations and there are m buffers. The customers enter through the first station and go through every stage until their process is finished in station n. Every machine consists of a single server and the process time in each station is t_s,i. There is K_i capacity in each station for the work in process: b_i places for customers in the queue plus the one being served.

What happens if all the available b places before a station i are occupied? The limitation on available space in a system will result in production shutdowns in the stations, which are then propagated backwards and forwards and whose effects increase the longer said shutdowns last.

For the station before machine i there is a phenomenon called blockage; this phenomenon is the fact that a customer cannot go on to the next stage in their process because there are is no place available in buffer j. This blockage is propagated to all the stations before machine i generating an accumulation of material waiting to be processed that could cause the line to totally shut down (Figure 2).

Figure 2.

Buffer full and propagation of blocking

Likewise, any lack of available material for the manufacturing process because of constraints of space combined with a slow machine or station results in a phenomenon called starvation, where the machines do not have any material to process and remain idle, and this is propagated forward (Figure 3).

Figure 3.

Propagation of blocking

There are two blocking mechanisms.

Blocking before service (Figure 4). When the buffer of station i+1 does not have any room for a new piece, then a signal is sent not to start processing the piece in station i.

Figure 4.

Blocking before service; The item cannot enter to the machine.