13.1 Introduction We all spend time waiting in lines (queues) – line at a bank, at a supermarket, at a fast-food restaurant, at a stoplight, and so on Televisions at a television repair shop, other than the one(s) being repaired, are essentially waiting in line to be repaired Messages are sent through a computer network, they often must wait in a queue before being processed.
13.1 Introduction cont The same type of analysis applies to all of these. The purpose of such an analysis is generally two fold: 1.We want to examine an existing system to quantify its operating characteristics. 2.We want to learn how to make a system better.
1.We want to examine an existing system to quantify its operating characteristics. – For example, if a fast-food restaurant currently employs 12 people in various jobs, the manager might be interested in determining the amount of time a typical customer must wait in line or how many customers are typically waiting in line Introduction cont
2.We want to learn how to make a system better. – The manager might find, for example, that the fast-food restaurant would do better, from an economic standpoint, by employing only 10 workers and deploying them in a different manner Introduction cont
The two basic modelling approaches are analytical and simulation Introduction cont
Analytical approach we search for mathematical formulas that describe the operating characteristics of the system, usually in steady state. The mathematical models are typically too complex to solve unless we make simplifying (and sometimes unrealistic) assumptions Introduction cont
Simulation allows us to analyze much more complex systems, without making many simplifying assumptions. However, the drawback to queueing simulation is that it usually requires specialized software packages or trained computer programmers to implement Introduction cont
We discuss several well-known queueing models that describe situations in the real world. These models illustrate how to calculate such operating characteristics as the average waiting time per customer, the average number of customers in line, and the fraction of time servers are busy. These analytical models generally require simplifying assumptions, and even then they can be difficult to understand Introduction cont
The inputs are typically: – mean customer arrival rates and – mean service times. The required outputs are typically: – mean waiting times in queues, – mean queue lengths, – the fraction of time servers are busy, and – possibly others Introduction cont
13.2 ELEMENTS OF QUEUEING MODELS Almost all queueing systems are alike in that customers 1.enter a system, 2.possibly wait in one or more queues, 3.get served, and then 4.depart.
Characteristics of Arrivals The arrival process consist of: 1.the timing of arrivals a well as 2.the types of arrivals
Characteristics of Arrivals cont Timing of arrivals Interarrival times - the times between successive customer arrivals These interarrival times might be known or nonrandom (doctors offices) or they can be unknown or random (Shoprite)
Types of arrivals: There are at least two issues. 1.Do customers arrive one at a time or in batches – carloads, for example? – The simplest system is when customer arrive one at a time, as we assume in all of the models in this chapter. Characteristics of Arrivals cont
2.Are all customers essentially alike, or can they be separated into priority classes? – At a computer centre, for example, certain jobs might receive higher priority and run first, whereas the lower- priority jobs might be sent to the back of the line and run only after midnight. – We assume throughout this chapter that all customers have the same priority. Characteristics of Arrivals cont
Another issue is whether (or how long) customers will wait in line. Balking - A customer might arrive to the system, see that too many customers are waiting in line, and decide not to enter the system at all. Characteristics of Arrivals cont
limited waiting room system – we assume there is a waiting room size so that if the number of customers in the system equals the waiting room size, newly arriving customers are not allowed to enter the system. – when the choice is made by the system, not the customer. Characteristics of Arrivals cont
Reneging - when a customer already in line becomes impatient and leaves the system before starting service Characteristics of Arrivals cont
Service Discipline When customers enter the system, they might have to wait in line until a server becomes available The service discipline is the rule that states which customer, from all who are waiting, goes into service next.
Service Discipline cont The most common service discipline is first- come-first-served (FCFS), – where customers are served in the order of their arrival. – All of the models we discuss use the FCFS discipline.
Other service disciplines are possible, including: – service-in-random-order (SRO), – last- come-first-served (LCFS) Service Discipline cont
One other aspect of the waiting process is whether there is a single line or multiple lines. 1.For example, most banks now have a single line. An arriving customer joins the end of the line. When any teller finishes service, the customer at the head of the line goes to that teller. 2.In contrast, most supermarkets have multiple lines. When a customer goes to a checkout counter, she must choose which of several lines to enter. Service Discipline cont
Service Characteristics Each customer is served by exactly one server, even when the system contains multiple servers. For example, when you enter a bank, you are eventually served by a single teller, even though several tellers are working. Interarrival times must typically be estimated from service time data in real applications.
Service Characteristics cont like a typical bank, where customers join a single line and are then served by the first available teller, we say the servers (tellers) are in parallel (see Figure 14.1).
Short-Run versus Steady-State Behaviour If you run a fast-food restaurant, you are particularly interested in the queueing behaviour during your peak lunchtime period. The customer arrival rate during this period increases sharply, and you probably employ more workers to meet the increased customer load. In this case, your primary interest is in the short-run behaviour of the system
Short-Run versus Steady-State Behaviour cont Analytical models are best suited for studying long-run behaviour. This type of analysis is called steady-state analysis and is the focus of much of the chapter. One requirement for steady-state analysis is that the parameters of the system remain constant for the entire time period.
Another requirement for steady-state analysis is that the system must be stable. This means that the servers must serve fast enough to keep up with arrivalsotherwise, the queue can theoretically grow without limit Short-Run versus Steady-State Behaviour cont
For example, in a single-server system where all arriving customers join the system, the requirement for system stability is that the arrival rate must be less than the service rate. If the system is not stable, the analytical models discussed in this chapter cannot be used. Short-Run versus Steady-State Behaviour cont
13.4 IMPORTANT QUEUEING RELATIONSHIP We typically calculate two general types of outputs in a queueing model: time averages and customer averages. Typical time averages are – L, the expected number of customers in the system – L Q, the expected number of customers in the queue – L S, the expected number of customers in service – P(all idle), the probability that all servers are idle – P(all busy), the probability that all servers are busy
13.4 IMPORTANT QUEUEING RELATIONSHIP cont Typical customer averages are: W, the expected time spent in the system (waiting in line or being served) W Q, the expected time spent in the queue W s, the expected time spent in service
Littles Formula λ = arrival rate (mean number of arrivals per time period) μ = service rate (mean number of people or items served per time period) U = server utilization (the long-run fraction of time the server is busy)
13.5 ANALYTICAL STEADY-STATE QUEUEING MODELS The Basic Single-Server Model (M/M/1) Kendell Notation: M/M/1 – first M implies that the distribution of interarrival times is exponential. – The second M implies that the distribution of service times is also exponential. – the 1 implies that there is a single server
The Basic Single-Server Model (M/M/1) Mean time between arrivals = 1/ λ The mean service time per customer = 1/ μ ρ = traffic intensity = λ/μ
The Basic Single-Server Model (M/M/1) cont ρ = traffic intensity = λ/μ This is called the traffic intensity, which is a very useful measure of the congestion of the system. In the system is stable only if ρ < 1. If ρ 1, so that λ μ, then arrivals occur at least as fast as the server can handle them; in the long run, the queue becomes infinitely largethat is, it is unstable. Therefore, we must assume that ρ < 1 to obtain steady-state results.
Assuming that the system is stable, let pn, be the steady-state probability that there are exactly n customer in the system (waiting in line or being served) at any point in time For example, p0 is the long-run fraction of time when there are no customers in the system, p1 is the long-run fraction of time when there is exactly one customer in the system, and so on. The Basic Single-Server Model (M/M/1) cont
The Basic Multi-Server Model (M/M/s) Many service facilities such as banks and postal branches employ multiple servers. These servers work in parallel, so that each customer goes to exactly one server for service and then departs
labelled the M/M/s model. First M means that interarrival times are exponentially distributed. The second M means that the service times for each server are exponentially distributed. the s in M/M/s denotes the number of servers. – E.G. M/M/3 - has 3 servers
Multiple-server facilities have two types of waiting line configurations. 1.The first, usually seen at supermarkets, is where each server has a separate line. Each customer must decide which line to join (and then either stay in that line or switch later on). 2.The second, seen at most banks and post offices, is where there is a single waiting line, from which customers are served in FCFS order. We examine only the second type because it is arguably the more common system in real-world situations and is much easier to analyze mathematically.
There are three inputs to this system: 1.the arrival rate λ, 2.the service rate (per server) μ, and the 3.number of servers s. To ensure that the system is stable, we must also assume that the traffic intensity, now given by ρ = λ /(sμ), is less than 1. – i.e.we require that the arrival rate λ be less than the maximum service rate sμ (which is achieved when all s servers are busy). – If the traffic intensity is not less than 1, the length of the queue eventually increases without bound.