Arrival. A customer or item entering a line or queue for service or further processing.

Arrival distribution. The variability in the number of arrivals per some selected period of time. This variability can be represented by selected probability distributions to best fit the particular situation, with the Poisson distribution being most commonly used.

Arrival rate. The average number of arrivals per some selected period of time. This continuous value can have a decimal component for small arrival rates, but it is often rounded to whole numbers for larger rates. This value is represented by the Greek lambda (λ) in waiting line equations. Some typical values are 30 products per day, 18 customers per hour, or 1.6 cars every 15 minutes.

Back office. A term that denotes those activities in a service process where direct contact with the customer is not required for their completion.

Balance equation. A useful condition in waiting line analysis that can be used to derive several performance measures. Consider a state n where there are n customers in the system and the queue is in a steady state. At this time, the inputs to state n are equal to the outputs from state n. That is, Rate(→ n) = Rate(← n). This allows the creation of a set of equations, one per state, which then can be mathematically manipulated to determine the probability of existence for each state. Given those probabilities, average line lengths and, subsequently, the average waiting times can be determined.

Balking. Refusal of a customer to enter a line the customer considers too long. It is a waiting line cost, particularly if the customer decides to never return. Balking is also an example of a state-dependent rate in which the probability of its occurrence is related to the length of the waiting line observed by the next arrival.

Birth and death process. A continuous Markov chain that is occasionally abbreviated by some authors as BDP. In queuing analysis, its state diagram can be used to determine how long it will take for a queuing situation to reach a steady state of behavior. When that condition exists, it can be used to derive the various waiting line performance formulas.

Blocking. Not allowing a customer to enter a line because of limited capacity. One example is when all the lines to a call center are busy. It is a waiting line cost.

Calling population. The pool of possible arrivals. For most customer services, the calling population is assumed to be infinite; but for some situations, this population has a limited or fixed size that affects subsequent arrival rates over time. Some may wonder why the adjective calling is used; the likely reason is because early waiting line analysis was focused on the needs of telephone companies to provide their users (callers) acceptable service at the lowest cost.

Central Limit Theorem. A law that states that the distribution of sample averages approaches that of a normal distribution as the number of samples increases. This occurs regardless of the nature of the underlying distribution of the individual data values from which the samples were taken.

Channel. A single server for a waiting line.

Coefficient of variation. The standard deviation/mean ratio for a distribution. This ratio is a key component of the Pollaczek-Khintchine formula used for general distributions in waiting line analysis equations. See also Pollaczek-Khintchine (P-K) formula.

Coxian distribution.¹ A special case of a phase-type distribution where the requirement for entering the first phase in a sequence of phases is relaxed so that the sequence can be entered at any phase. A typical application would be sequencing analysis in a job shop.

Dequeue (or sometimes deque). A concept usually reserved for computer applications, where a customer can be added or subtracted from either the head or the tail of the line.

Deterministic. Having a predictable value with a very narrow range of variance. As a result, deterministic variables do not require probability distributions to describe their behavior in analysis methods and can usually be represented by a constant value. In the context of this monograph, some examples are the service time for an automated process, arrival times created by a production schedule, the number of servers, finite population size, and finite capacity. See also stochastic.

Erlang distribution.² A special case of the gamma distribution where the shape factor k is restricted to integer values. The shape factor represents the number of independent exponential distributions added together to form the distribution. When k = 1, the distribution defaults to a single exponential distribution.

Erlang loss function. Also referred to as the Erlang B expression, can be used to estimate how many customers are turned away (blocked) by limited capacity (no waiting positions, e.g., all lines are busy at a call center). The Erlang C expression, which is more complicated, assumes that there are infinite waiting positions. It is used to estimate the probability that a customer will arrive, find that all servers are currently busy, and then get into line to wait until a server becomes available.

FCFS. First come, first serve. The most common priority rule for a waiting line.

FIFO. First in, first out. Same as FCFS.

Front office. A term used to denote those activities in a service process where direct contact with a customer is necessary for their completion.

Interarrival distribution. The variability in the interarrival time, which can be represented by selected probability distributions that best fit the particular situation, with the exponential distribution being the most commonly used.

Interarrival time. The time between arrivals in a waiting line. The average value is the inverse of the arrival rate (i.e., 1/λ). The sequence of interarrival times is often characterized as a Markovian process.

Jockeying. Changing from a slowly moving line to another line that is perceived to be moving faster in a multiple-channel situation using separate lines for each server.

Kendall notation.³ A shorthand method for identifying different waiting line models using designated symbols for the arrival rate distribution, the service time distribution, the number of channels, the line length limitation, the calling population size, and the priority rule.

Line length. The number of customers or items in a queue. The average line length is characterized by two continuous values: the average number in line (L_q) awaiting service and the total number in the system (L), which includes those being served.

Little’s Law. An observation that the ratio between given average line length and waiting time values is characterized by the average arrival rate value for all waiting line models (λ = L/W).

Makespan. The total time required to complete a group of jobs in manufacturing. For waiting lines, this corresponds to the total amount of time required to process a group of waiting customers.

Markovian process.⁴ A process characterized by a sequence of random variables (x₁, x₂, x₃, . . .) whose order is indicated by increasing values of a parameter, usually time in waiting line scenarios, and having the property that any prediction of the next value of the sequence (x_n) can be based on knowing just the current state or previous value (x_n−1). That is, the future value of such a variable is independent of the values of all of the previous values but the last one.

Memoryless. A condition where a value is independent of any previous value. That is, it can be said that the value has no memory of any preceding values.

PASTA. An abbreviation for the statement “Poisson arrivals see time averages.” In other words, for any line with exponentially distributed interarrival times, each customer has the same probability of seeing a given line condition. Therefore, the average of what a frequent customer observes will be the same as the average determined by waiting line analysis.

Phase. A single step for a service with no intervening waiting lines.

Phase-type distribution. A distribution for one or more phases in sequence whose intervening arrival rates can be characterized by interrelated Poisson distributions. These complex expressions are beyond the scope of this monograph and are mentioned here only for readers wishing to delve further into analytical solutions.

Pollaczek-Khintchine (P-K) formula.⁵ An expression developed to determine the average delay in a single-channel waiting line for any general service time distribution. All that is needed to be known about the distribution is its mean and variance. The generalized formula in the context of the terms used in this monograph is as follows:

POS. Point of sale. Useful information regarding the nature of services performed and the types of customers served can be gained by a business analyzing its POS data.

Priority queue. A waiting line where the order in which arrivals are served is different from first come, first served. Different preemptive and nonpreemptive rules can be applied within a single line, or separate lines can be established for different priorities.

Priority rule. A waiting line discipline for serving arrivals after they enter the queue. The most commonly used rule is first come, first served; but in other more urgent situations, the most important needs will prevail, such as in emergency room queues. See also priority queue.

Probability of zero customers. The percentage of time when a service business will have no customers in line and being served. This value is represented by the symbol P₀ in most waiting line formulas and is important to managers because it allows an estimate of how much time servers can be available to work on other activities not directly related to customer service.

Queue. A line of people waiting for some service or a sequence of items waiting for the next process step. This term is more widely used outside the United States and is considered synonymous with the term waiting line.

Reneging. Leaving a line before being served after having spent some time in it. If a customer is dissatisfied enough to not come back, it is a significant waiting line cost.

Service blueprint. A process diagram for services where the activities are separated into two groups: customer involvement required (front office) and support behind the scenes (back office). Sometimes the service blueprint adds a third group of activities, where customer involvement may or may not be required depending on the particular circumstances (such as when a credit card is rejected for payment).

Service distribution. The variability in a service time, which can be represented by selected probability distributions that best fit the particular situation, with the exponential distribution being most commonly used.

Service rate. The average number of arrivals serviced or processed per some selected period of time. This continuous value can have a decimal component for small service rates, but it is normally rounded to whole numbers for larger rates. This value is represented by the Greek mu (µ) in waiting line equations. Some typical values are 1.2 customers per minute, 6 cars per hour, or 120 products per week.

Service time. The time required to perform a service or process. The average value is the inverse of the service rate (i.e., 1/µ).

Sojourn time. The total time spent in a service system. See also waiting time.

State-dependent rate. A rate influenced by the current state of a waiting line. See also balking and reneging for examples of factors creating a state-dependent arrival rate. Some models that have a state-dependent arrival rate are those dealing with a limited population (K) or limited queuing capacity (N).

State diagram. A graphical method showing the conditions for each state of a waiting line. Each state represents a specific number of persons or items in the queue. When the queue reaches a steady-state condition, the movements from one state to another are governed by a balance equation.

Stochastic. Having an unpredictable value because of the possibility of a wide range of possible results. As a result, stochastic variables must be represented by either discrete or continuous probability distributions in analysis methods such as queuing equations. Examples in the context of this monograph are infinite population arrival times and service times, how long a customer is willing to wait before giving up on a service line, and which of the several service lines available is selected by the next customer. See also deterministic.

Utilization factor. The ratio of the arrival rate λ to the service rate µ. This value is represented by the Greek rho (ρ = λ/u) in waiting line equations. This is analogous to machine capacity reduced by some percentage (cushion) to allow some time for preventive maintenance or reserve some capacity for unexpected increases in demand. This factor must be less than 1 for a single-channel waiting line situation to reach a steady state. Modifications of this factor to accommodate conditions associated with multiple servers and/or limited calling populations are indicated in this monograph by the use of appropriate subscript notation as defined in the text and in appendix B.

Waiting line cost. A measure of the estimated level of overall customer dissatisfaction. This cost can take the form of lost profit when customers are turned away by a busy situation, estimates of lost business caused by dissatisfied customers, or even work required to correct a service error. See also balking, blocking, and reneging.

Waiting time. The time spent in a waiting line or queue. The average waiting time is characterized by two values: the average time (W_q) spent in line awaiting service and the total time (W) spent in the system (waiting in line plus service time). W is often called the throughput time in manufacturing applications and the sojourn time in service applications.

WIP. Work in process for a manufacturing line. It is the number of items in a manufacturing system waiting for processing or being worked on.