The main types of queuing networks are open, closed, and mixed networks, classified by how customers or jobs enter and leave the system. Networks can also be described by their layout, routing rules, and service-station design, so these labels are different ways of classifying the same model rather than one exclusive list.
What a Queuing Network Represents
A queuing network is a model of connected service stations through which customers, jobs, messages, or other units move. A station usually consists of a queue and one or more servers. A server might represent a bank teller, a processor, a machine, a nurse, or a communication link. Units may wait when all servers are busy, receive service, and then move to another station or leave the network.
The network view is useful when a delay depends on more than one step. A computer request, for example, might wait for a processor, access a disk, and return to the processor. Modeling each queue separately can miss the effect of jobs circulating between them. A network model captures both the local waiting at each station and the movement between stations.
To describe a network, analysts specify its stations, arrival patterns, service times, number of servers, queue capacities, and routing probabilities. The assumptions matter: two systems with the same diagram can behave differently if one has variable service times, limited waiting space, or feedback between stations.
Open, Closed, and Mixed Networks
The most common high-level distinction concerns the customer population and its boundary with the outside world.
- Open networks: Customers arrive from outside, pass through one or more stations, and eventually depart. The number of customers inside is not fixed. A help desk receiving new calls is a simple example. In a stable open model, the long-run arrival rate must be low enough relative to service capacity for queues not to grow without bound.
- Closed networks: A fixed population circulates among stations, with no external arrivals or departures. When one customer finishes at a station, it moves elsewhere in the network. A model of a fixed number of terminals repeatedly submitting work to a computer system is a classic example. The population size affects throughput and waiting times.
- Mixed networks: These combine open and closed customer flows. Some jobs arrive from outside and leave after service, while another population circulates within the system. A manufacturing model could include circulating reusable containers alongside new production orders entering and completed orders departing.
These categories are not descriptions of physical layout. An open network can be a straight line, a branching system, or a network with loops. Likewise, a closed network can contain several stations and many possible routes. The terms tell you whether the modeled population crosses the system boundary, not how the stations are arranged.
Common Network Layouts
Another useful classification describes the network’s topology: which stations connect to which others. A tandem or series network sends units through stations in sequence. For instance, an item might pass through inspection, repair, and final testing. The queue at an early station can affect the flow reaching later stations, so each step’s capacity matters.
In a parallel layout, customers can be directed to one of several service stations. A supermarket checkout area or a bank with multiple tellers is a familiar analogy. The model must specify how jobs are assigned: they may join a common line, choose a particular server, or be routed according to a rule such as selecting the shortest queue. Those alternatives can produce different waiting times even when the number of servers is the same.
A feedforward network has routes that move in one general direction without returning to earlier stations. A feedback network includes loops, so a job may revisit a station. In a repair process, an item that fails a test might return for another adjustment. A branching network splits work among different paths; paths may later merge, or some jobs may leave after a branch. Real systems often combine these patterns rather than matching a single neat diagram.
Service Stations and Queue Structures
Networks can also be grouped by the service resources at each station. A single-server station has one active server, while a multi-server station has several servers able to work at once. The distinction is important when estimating capacity: adding parallel servers can reduce delay, but the benefit depends on how arrivals are distributed and whether all servers share one queue.
Queue capacity is another design feature. A station with unlimited modeled capacity can keep accepting waiting jobs, while a finite-capacity station can reject arrivals or block upstream work when full. Many textbook models assume infinite buffers because they simplify analysis. In a real production line, however, finite space between machines may cause blocking: a completed unit cannot move forward, leaving the upstream machine occupied.
Service discipline describes the order in which waiting jobs receive service. First-come, first-served is common, but priority rules, processor sharing, and last-come, first-served can also be modeled. A network may use different disciplines at different stations. The discipline can change which customers wait longest, even if the average workload and service capacity stay constant.
Routing Rules and Customer Movement
Routing determines where a customer goes after completing service. In a probabilistic routing model, each possible next station is assigned a probability. For example, after a consultation, some patients may go to a laboratory, some to imaging, and others may leave. The probabilities for all possible next destinations, including departure, must account for the full set of routes from that station.
Routing can instead depend on a job’s type, its previous history, or the current state of the system. A dispatcher might send work to the least-loaded machine, or a failed test might trigger rework. Such state-dependent rules can make analysis more difficult because the route is no longer described by fixed probabilities alone.
It is also important to distinguish a physical network from a mathematical routing diagram. A real customer may make choices, abandon a queue, or require several visits. If those behaviors matter to the question being studied, they need to be represented in the model rather than assumed away.
Analytical Model Families
Some network models have special mathematical properties that allow performance measures to be calculated from station-level information. Jackson networks are a well-known class of open networks with assumptions such as Poisson external arrivals, exponential service times, and suitable probabilistic routing. Under the relevant conditions, the network has a product-form steady-state distribution: its joint state probabilities can be expressed using factors associated with individual stations.
Gordon–Newell networks are a classic product-form class for closed networks with a fixed customer population and particular service assumptions. These results are valuable because they make some complex systems tractable, but they do not apply automatically to every network. General service-time distributions, priorities, blocking, state-dependent routing, or other features may require different methods.
When exact formulas are unavailable or the system is highly detailed, analysts may use simulation. Discrete-event simulation represents arrivals, service completions, and routing events over time. It can handle many practical complications, but its output is an estimate based on the model and simulation runs; it does not remove the need to validate assumptions or examine uncertainty.
What Network Measures Can Tell You
Common measures include throughput, average number of jobs at each station, queue length, waiting time, total time in the system, and server utilization. Throughput is the rate at which jobs complete a process. Utilization is the fraction of time a server is busy, under the chosen definition and observation period. These measures answer different questions: a high utilization may indicate efficient use of resources, but it can also leave little spare capacity when demand fluctuates.
For stable systems, Little’s law relates average number in a system, throughput, and average time in the system: average number equals throughput multiplied by average time. It is a general relationship under appropriate long-run conditions, not a formula that by itself predicts performance. It can help check whether measured or modeled quantities are mutually consistent.
How to Choose a Network Type
Start with the decision you need to make, then select the simplest model that represents the important features. A model intended to compare staffing plans may not need the same detail as one examining production blocking or software contention. Before choosing a model family, map the system boundary and its routes.
- Decide whether jobs arrive from outside, circulate in a fixed population, or include both kinds of flow.
- Identify each meaningful service station, its number of servers, service capacity, and waiting-space limits.
- Record where jobs go after service, including departures, branching, rework, and abandonment if relevant.
- Check whether arrivals and service times can reasonably be represented by the distributions required by an analytical model.
- Choose an analytical method or simulation based on the model’s complexity and the accuracy needed for the decision.
Use operational data where possible to estimate arrival rates, service times, routing shares, and variability. Then compare model predictions with observed behavior. If measured queues are consistently longer than predicted, a missing route, peak-demand pattern, downtime, or finite buffer may explain the gap. A mathematically elegant model is not useful if it omits the mechanism driving the real delay.
Example: A Service Process With Rework
Consider a device-repair operation with intake, diagnosis, repair, and testing stations. New devices arrive from outside and completed devices leave, so it is an open network. The main route is a series of stations, but testing creates a feedback loop because some devices return to repair. If diagnosis can send devices to either of two specialist repair teams, the network also branches.
This example shows why a single label is not enough. Calling the operation an open network says something about arrivals and departures; calling it a feedback network describes a route; calling a repair team multi-server describes its service capacity. To estimate turnaround time, the model needs the share of devices sent back for rework, service-time variation, and the number of repairers. If the waiting area has limited space, capacity constraints may matter too.
Frequently Asked Questions
What are the three main types of queuing networks?
They are open, closed, and mixed networks. Open networks accept outside arrivals and allow departures; closed networks keep a fixed population circulating; mixed networks include both external and circulating populations.
Is a tandem network the same as an open network?
No. Tandem describes a sequence of stations, while open describes whether customers enter from and leave to the outside. A tandem network can be open, closed, or part of a mixed model.
What is the difference between a queue and a queuing network?
A queue is a waiting line associated with a service facility. A queuing network connects multiple queues or service stations and represents how jobs move among them, including any routes that lead back to earlier stations.
Which type of queuing network is best?
There is no universally best type. The correct model is the one that reflects the system being studied. Use an open model for external arrivals and departures, a closed model for a fixed circulating population, and a mixed model when both occur.
Are Jackson networks the same as open networks?
No. Jackson networks are a particular analytically tractable class of open networks with specific assumptions about arrivals, service, and routing. Many open networks do not meet those assumptions and need another analytical approach or simulation.