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A Basic Queueing System. Herr Cutter’s Barber Shop. Herr Cutter is a German barber who runs a one-man barber shop. Herr Cutter opens his shop at 8:00 A.M. The table shows his queueing system in action over a typical morning. Arrivals.

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herr cutter s barber shop
Herr Cutter’s Barber Shop
  • Herr Cutter is a German barber who runs a one-man barber shop.
  • Herr Cutter opens his shop at 8:00 A.M.
  • The table shows his queueing system in action over a typical morning.
arrivals
Arrivals
  • The time between consecutive arrivals to a queueing system are called the interarrival times.
  • The expected number of arrivals per unit time is referred to as the mean arrival rate.
  • The symbol used for the mean arrival rate is

l = Mean arrival rate for customers coming to the queueing system

where l is the Greek letter lambda.

  • The mean of the probability distribution of interarrival times is

1 / l = Mean interarrival time

  • Most queueing models assume that the form of the probability distribution of interarrival times is an exponential distribution.
properties of the exponential distribution
Properties of the Exponential Distribution
  • There is a high likelihood of small interarrival times, but a small chance of a very large interarrival time. This is characteristic of interarrival times in practice.
  • For most queueing systems, the servers have no control over when customers will arrive. Customers generally arrive randomly.
  • Having random arrivals means that interarrival times are completely unpredictable, in the sense that the chance of an arrival in the next minute is always just the same.
  • The only probability distribution with this property of random arrivals is the exponential distribution.
  • The fact that the probability of an arrival in the next minute is completely uninfluenced by when the last arrival occurred is called the lack-of-memory property.
the queue
The Queue
  • The number of customers in the queue (or queue size) is the number of customers waiting for service to begin.
  • The number of customers in the system is the number in the queue plus the number currently being served.
  • The queue capacity is the maximum number of customers that can be held in the queue.
  • An infinite queue is one in which, for all practical purposes, an unlimited number of customers can be held there.
  • When the capacity is small enough that it needs to be taken into account, then the queue is called a finite queue.
  • The queue discipline refers to the order in which members of the queue are selected to begin service.
    • The most common is first-come, first-served (FCFS).
    • Other possibilities include random selection, some priority procedure, or even last-come, first-served.
service
Service
  • When a customer enters service, the elapsed time from the beginning to the end of the service is referred to as the service time.
  • Basic queueing models assume that the service time has a particular probability distribution.
  • The symbol used for the mean of the service time distribution is 1 / m = Mean service timewhere m is the Greek letter mu.
  • The interpretation of m itself is the mean service rate.m = Expected service completions per unit time for a single busy server
some service time distributions
Some Service-Time Distributions
  • Exponential Distribution
    • The most popular choice.
    • Much easier to analyze than any other.
    • Although it provides a good fit for interarrival times, this is much less true for service times.
    • Provides a better fit when the service provided is random than if it involves a fixed set of tasks.
    • Standard deviation: s = Mean
  • Constant Service Times
    • A better fit for systems that involve a fixed set of tasks.
    • Standard deviation: s = 0.
  • Erlang Distribution
    • Fills the middle ground between the exponential distribution and constant.
    • Has a shape parameter, k that determines the standard deviation.
    • In particular, s = mean / (k)
labels for queueing models
Labels for Queueing Models

To identify which probability distribution is being assumed for service times (and for interarrival times), a queueing model conventionally is labeled as follows: Distribution of service times — / — / — Number of Servers Distribution of interarrival times

The symbols used for the possible distributions areM = Exponential distribution (Markovian)D = Degenerate distribution (constant times)Ek = Erlang distribution (shape parameter = k)GI = General independent interarrival-time distribution (any distribution)G = General service-time distribution (any arbitrary distribution)

summary of usual model assumptions
Summary of Usual Model Assumptions
  • Interarrival times are independent and identically distributed according to a specified probability distribution.
  • All arriving customers enter the queueing system and remain there until service has been completed.
  • The queueing system has a single infinite queue, so that the queue will hold an unlimited number of customers (for all practical purposes).
  • The queue discipline is first-come, first-served.
  • The queueing system has a specified number of servers, where each server is capable of serving any of the customers.
  • Each customer is served individually by any one of the servers.
  • Service times are independent and identically distributed according to a specified probability distribution.
choosing a measure of performance
Choosing a Measure of Performance
  • Managers who oversee queueing systems are mainly concerned with two measures of performance:
    • How many customers typically are waiting in the queueing system?
    • How long do these customers typically have to wait?
  • When customers are internal to the organization, the first measure tends to be more important.
    • Having such customers wait causes lost productivity.
  • Commercial service systems tend to place greater importance on the second measure.
    • Outside customers are typically more concerned with how long they have to wait than with how many customers are there.
defining the measures of performance
Defining the Measures of Performance

L = Expected number of customers in the system, including those being served (the symbol L comes from Line Length).

Lq = Expected number of customers in the queue, which excludes customers being served.

W = Expected waiting time in the system (including service time) for an individual customer (the symbol W comes from Waiting time).

Wq = Expected waiting time in the queue (excludes service time) for an individual customer.

These definitions assume that the queueing system is in a steady-state condition.

relationship between l w l q and w q
Relationship between L, W, Lq, and Wq
  • Little’s formula states thatL = lWandLq= lWq
  • Since 1/m is the expected service timeW = Wq+ 1/m
  • Combining the above relationships leads toL = Lq + l/m
using probabilities as measures of performance
Using Probabilities as Measures of Performance
  • In addition to knowing what happens on the average, we may also be interested in worst-case scenarios.
    • What will be the maximum number of customers in the system? (Exceeded no more than, say, 5% of the time.)
    • What will be the maximum waiting time of customers in the system? (Exceeded no more than, say, 5% of the time.)
  • Statistics that are helpful to answer these types of questions are available for some queueing systems:
    • Pn = Steady-state probability of having exactly n customers in the system.
    • P(W ≤ t) = Probability the time spent in the system will be no more than t.
    • P(Wq ≤ t) = Probability the wait time will be no more than t.
  • Examples of common goals:
    • No more than three customers 95% of the time: P0 + P1 + P2 + P3 ≥ 0.95
    • No more than 5% of customers wait more than 2 hours: P(W ≤ 2 hours) ≥ 0.95
the dupit corp problem
The Dupit Corp. Problem
  • The Dupit Corporation is a longtime leader in the office photocopier marketplace.
  • Dupit’s service division is responsible for providing support to the customers by promptly repairing the machines when needed. This is done by the company’s service technical representatives, or tech reps.
  • Current policy: Each tech rep’s territory is assigned enough machines so that the tech rep will be active repairing machines (or traveling to the site) 75% of the time.
    • A repair call averages 2 hours, so this corresponds to 3 repair calls per day.
    • Machines average 50 workdays between repairs, so assign 150 machines per rep.
  • Proposed New Service Standard: The average waiting time before a tech rep begins the trip to the customer site should not exceed two hours.
alternative approaches to the problem
Alternative Approaches to the Problem
  • Approach Suggested by John Phixitt: Modify the current policy by decreasing the percentage of time that tech reps are expected to be repairing machines.
  • Approach Suggested by the Vice President for Engineering: Provide new equipment to tech reps that would reduce the time required for repairs.
  • Approach Suggested by the Chief Financial Officer: Replace the current one-person tech rep territories by larger territories served by multiple tech reps.
  • Approach Suggested by the Vice President for Marketing: Give owners of the new printer-copier priority for receiving repairs over the company’s other customers.
the queueing system for each tech rep
The Queueing System for Each Tech Rep
  • The customers: The machines needing repair.
  • Customer arrivals: The calls to the tech rep requesting repairs.
  • The queue: The machines waiting for repair to begin at their sites.
  • The server: The tech rep.
  • Service time: The total time the tech rep is tied up with a machine, either traveling to the machine site or repairing the machine. (Thus, a machine is viewed as leaving the queue and entering service when the tech rep begins the trip to the machine site.)
notation for single server queueing models
Notation for Single-Server Queueing Models
  • l = Mean arrival rate for customers = Expected number of arrivals per unit time1/l = expected interarrival time
  • m = Mean service rate (for a continuously busy server) = Expected number of service completions per unit time1/m = expected service time
  • r = the utilizationfactor= the average fraction of time that a server is busy serving customers = l / m
the m m 1 model
The M/M/1 Model
  • Assumptions
    • Interarrival times have an exponential distribution with a mean of 1/l.
    • Service times have an exponential distribution with a mean of 1/m.
    • The queueing system has one server.
  • The expected number of customers in the system is

L = r / (1 –r) = l / (m– l)

  • The expected waiting time in the system is

W = (1 / l)L = 1 / (m – l)

  • The expected waiting time in the queue is

Wq= W – 1/m = l / [m(m – l)]

  • The expected number of customers in the queue is

Lq= lWq = l2 / [m (m – l)]

the m m 1 model1
The M/M/1 Model
  • Theprobability of having exactly n customers in the system is

Pn = (1 – r)rnThus,P0 = 1 – rP1 = (1 – r)rP2 = (1 – r)r2 : :

  • The probability that the waiting time in the system exceeds t is

P(W > t) = e–m (1–r)tfor t ≥ 0

  • The probability that the waiting time in the queue exceeds t is

P(Wq > t) = r e–m (1–r)tfor t ≥ 0

john phixitt s approach reduce machines rep
John Phixitt’s Approach (Reduce Machines/Rep)
  • The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq≤ 1/4 day).
  • John Phixitt’s suggested approach is to lower the tech rep’s utilization factor sufficiently to meet the new service requirement. Lower r = l / m, until Wq≤ 1/4 day,wherel = (Number of machines assigned to tech rep) / 50.
the m g 1 model
The M/G/1 Model
  • Assumptions
    • Interarrival times have an exponential distribution with a mean of 1/l.
    • Service times (T) can have any probability distribution.

E(T) = 1/m , Var(T) = s2.

3. The queueing system has one server.

  • The probability of zero customers in the system is

P0 = 1 – r

  • The expected number of customers in the queue is

Lq= l2[Var(T)+ E(T)2] / [2(1 – lE(T))]

  • The expected number of customers in the system is

L = Lq + l/m

The expected waiting time in the queue is

Wq= Lq/ l

  • The expected waiting time in the system is

W = Wq+ 1/m

the values of s and l q for the m g 1 model with various service time distributions
The Values of s and Lqfor the M/G/1 Modelwith Various Service-Time Distributions
  • The expected number of customers in the queue is

Lq= l2[Var(T)+ E(T)2] / [2(1 – lE(T))]=[l2s2 + r2] / [2(1 – r)]

vp for engineering approach new equipment
VP for Engineering Approach (New Equipment)
  • The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq≤ 1/4 day).
  • The Vice President for Engineering has suggested providing tech reps with new state-of-the-art equipment that would reduce the time required for the longer repairs.
  • After gathering more information, they estimate the new equipment would have the following effect on the service-time distribution:
    • Decrease the mean from 1/4 day to 1/5 day.
    • Decrease the standard deviation from 1/4 day to 1/10 day.
the m m s model
The M/M/s Model
  • Assumptions
    • Interarrival times have an exponential distribution with a mean of 1/l.
    • Service times have an exponential distribution with a mean of 1/m.
    • Any number of servers (denoted by s).
  • With multiple servers, the formula for the utilization factor becomesr = l / smbut still represents that average fraction of time that individual servers are busy.
cfo suggested approach combine into teams
CFO Suggested Approach (Combine Into Teams)
  • The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq≤ 1/4 day).
  • The Chief Financial Officer has suggested combining the current one-person tech rep territories into larger territories that would be served jointly by multiple tech reps.
  • A territory with two tech reps:
    • Number of machines = 300 (versus 150 before)
    • Mean arrival rate = l = 6 (versus l = 3 before)
    • Mean service rate = m = 4 (as before)
    • Number of servers = s = 2 (versus s = 1 before)
    • Utilization factor = r = l/sm = 0.75 (as before)
cfo suggested approach teams of three
CFO Suggested Approach (Teams of Three)
  • The Chief Financial Officer has suggested combining the current one-person tech rep territories into larger territories that would be served jointly by multiple tech reps.
  • A territory with three tech reps:
    • Number of machines = 450 (versus 150 before)
    • Mean arrival rate = l = 9 (versus l = 3 before)
    • Mean service rate = m = 4 (as before)
    • Number of servers = s = 3 (versus s = 1 before)
    • Utilization factor = r = l/sm = 0.75 (as before)
the four approaches under considerations
The Four Approaches Under Considerations

Decision: Adopt the third proposal

some insights about designing queueing systems
Some Insights About Designing Queueing Systems
  • When designing a single-server queueing system, beware that giving a relatively high utilization factor (workload) to the server provides surprisingly poor performance for the system.
  • Decreasing the variability of service times (without any change in the mean) improves the performance of a queueing system substantially.
  • Multiple-server queueing systems can perform satisfactorily with somewhat higher utilization factors than can single-server queueing systems. For example, pooling servers by combining separate single-server queueing systems into one multiple-server queueing system greatly improves the measures of performance.
  • Applying priorities when selecting customers to begin service can greatly improve the measures of performance for high-priority customers.
economic analysis of the number of servers to provide
Economic Analysis of the Number of Servers to Provide
  • In many cases, the consequences of making customers wait can be expressed as a waiting cost.
  • The manager is interested in minimizing the total cost. TC = Expected total cost per unit time SC = Expected service cost per unit time WC = Expected waiting cost per unit timeThe objective is then to choose the number of servers so as to Minimize TC = SC + WC
  • When each server costs the same (Cs= cost of server per unit time), SC = Cs s
  • When the waiting cost is proportional to the amount of waiting (Cw = waiting cost per unit time for each customer), WC = Cw L
acme machine shop
Acme Machine Shop
  • The Acme Machine Shop has a tool crib for storing tool required by shop mechanics.
  • Two clerks run the tool crib.
  • The estimates of the mean arrival rate l and the mean service rate (per server) m arel = 120 customers per hourm = 80 customers per hour
  • The total cost to the company of each tool crib clerk is $20/hour, so Cs= $20.
  • While mechanics are busy, their value to Acme is $48/hour, so Cw = $48.
  • Choose s so as to Minimize TC = $20s + $48L.