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Computer Science, Informatik 4 Communication and Distributed Systems
SimulationSimulation
Modeling and Performance Analysis with Discrete-Event Simulationg y
Dr. Mesut Güneş
Computer Science, Informatik 4 Communication and Distributed Systems
Chapter 8
Queueing Models
Computer Science, Informatik 4 Communication and Distributed Systems
ContentsContents
Characteristics of Queueing SystemsCharacteristics of Queueing SystemsQueueing Notation – Kendall NotationLong-run Measures of Performance of Queueing Systemsg g ySteady-state Behavior of Infinite-Population Markovian ModelsSteady-state Behavior of Finite-Population ModelsNetworks of Queues
Dr. Mesut GüneşChapter 8. Queueing Models 3
Computer Science, Informatik 4 Communication and Distributed Systems
PurposePurpose
Simulation is often used in the analysis of queueing models.Simulation is often used in the analysis of queueing models.A simple but typical queueing model
SW i i li
Calling population
Queueing models provide the analyst with a powerful tool for
ServerWaiting line
Queueing models provide the analyst with a powerful tool for designing and evaluating the performance of queueing systems.Typical measures of system performance
S tili ti l th f iti li d d l f t• Server utilization, length of waiting lines, and delays of customers• For relatively simple systems, compute mathematically• For realistic models of complex systems, simulation is usually required
Dr. Mesut GüneşChapter 8. Queueing Models 4
p y y q
Computer Science, Informatik 4 Communication and Distributed Systems
OutlineOutline
Discuss some well-known modelsDiscuss some well known models • Not development of queueing theory, for this see other class!
We will deal with• General characteristics of queues• Meanings and relationships of important performance measures• Estimation of mean measures of performance• Effect of varying input parameters• Mathematical solutions of some basic queueing models
Dr. Mesut GüneşChapter 8. Queueing Models 5
Computer Science, Informatik 4 Communication and Distributed Systems
Characteristics of Queueing SystemsCharacteristics of Queueing Systems
Dr. Mesut GüneşChapter 8. Queueing Models 6
Computer Science, Informatik 4 Communication and Distributed Systems
Characteristics of Queueing SystemsCharacteristics of Queueing Systems
Key elements of queueing systemsKey elements of queueing systems• Customer: refers to anything that arrives at a facility and requires
service, e.g., people, machines, trucks, emails.S f t th t id th t d i• Server: refers to any resource that provides the requested service, e.g., repairpersons, retrieval machines, runways at airport.
System Customers ServerSystem Customers ServerReception desk People ReceptionistHospital Patients NursesAirport Airplanes RunwayProduction line Cases Case-packerR d k C T ffi li hRoad network Cars Traffic lightGrocery Shoppers Checkout stationComputer Jobs CPU disk CD
Dr. Mesut GüneşChapter 8. Queueing Models 7
Computer Jobs CPU, disk, CDNetwork Packets Router
Computer Science, Informatik 4 Communication and Distributed Systems
Calling PopulationCalling Population
Calling population: the population of potential customers, may beCalling population: the population of potential customers, may be assumed to be finite or infinite.• Finite population model: if arrival rate depends on the number of
customers being served and waiting e g model of one corporate jet if itcustomers being served and waiting, e.g., model of one corporate jet, if it is being repaired, the repair arrival rate becomes zero.
n n-1
• Infinite population model: if arrival rate is not affected by the number of customers being served and waiting e g systems with large populationcustomers being served and waiting, e.g., systems with large population of potential customers.
∞Dr. Mesut GüneşChapter 8. Queueing Models 8
∞
Computer Science, Informatik 4 Communication and Distributed Systems
System CapacitySystem Capacity
System Capacity: a limit on the number of customers that maySystem Capacity: a limit on the number of customers that may be in the waiting line or system.• Limited capacity, e.g., an automatic car wash only has room for 10 cars
t it i li t t th h ito wait in line to enter the mechanism.
• Unlimited capacity e g concert ticket sales with no limit on the number
ServerWaiting line
• Unlimited capacity, e.g., concert ticket sales with no limit on the number of people allowed to wait to purchase tickets.
Dr. Mesut GüneşChapter 8. Queueing Models 9
ServerWaiting line
Computer Science, Informatik 4 Communication and Distributed Systems
Arrival ProcessArrival Process
For infinite-population models:For infinite population models:• In terms of interarrival times of successive customers.• Random arrivals: interarrival times usually characterized by a probability
distributiondistribution.- Most important model: Poisson arrival process (with rate λ), where An
represents the interarrival time between customer n-1 and customer n, and is exponentially distributed (with mean 1/λ)exponentially distributed (with mean 1/λ).
• Scheduled arrivals: interarrival times can be constant or constant plus or minus a small random amount to represent early or late arrivalsminus a small random amount to represent early or late arrivals.
- Example: patients to a physician or scheduled airline flight arrivals to an airport
• At least one customer is assumed to always be present, so the server is never idle, e.g., sufficient raw material for a machine.
Dr. Mesut GüneşChapter 8. Queueing Models 10
Computer Science, Informatik 4 Communication and Distributed Systems
Arrival ProcessArrival Process
For finite-population models:For finite population models:• Customer is pending when the customer is outside the queueing system,
e.g., machine-repair problem: a machine is “pending” when it is operating it becomes “not pending” the instant it demands service fromoperating, it becomes not pending the instant it demands service from the repairman.
• Runtime of a customer is the length of time from departure from the queueing system until that customer’s next arrival to the queue e gqueueing system until that customer s next arrival to the queue, e.g., machine-repair problem, machines are customers and a runtime is time to failure (TTF).
• Let A (i) A (i) be the successive runtimes of customer i and S (i) S (i)• Let A1(i), A2
(i), … be the successive runtimes of customer i, and S1(i), S2
(i)
be the corresponding successive system times:
Dr. Mesut GüneşChapter 8. Queueing Models 11
Computer Science, Informatik 4 Communication and Distributed Systems
Queue Behavior and Queue DisciplineQueue Behavior and Queue Discipline
Queue behavior: the actions of customers while in a queueQueue behavior: the actions of customers while in a queue waiting for service to begin, for example:• Balk: leave when they see that the line is too long
R l ft b i i th li h it i t l l• Renege: leave after being in the line when its moving too slowly• Jockey: move from one line to a shorter line
Queue discipline: the logical ordering of customers in a queue that determines which customer is chosen for service when a server becomes free, for example:• First-in-first-out (FIFO)• Last-in-first-out (LIFO)Last-in-first-out (LIFO)• Service in random order (SIRO)• Shortest processing time first (SPT)
Dr. Mesut GüneşChapter 8. Queueing Models 12
• Service according to priority (PR)
Computer Science, Informatik 4 Communication and Distributed Systems
Service Times and Service MechanismService Times and Service Mechanism
Service times of successive arrivals are denoted by S1, S2, S3.Service times of successive arrivals are denoted by S1, S2, S3.• May be constant or random.• {S1, S2, S3, …} is usually characterized as a sequence of independent and
identically distributed random variables e g exponential Weibullidentically distributed random variables, e.g., exponential, Weibull, gamma, lognormal, and truncated normal distribution.
A queueing system consists of a number of service centers and interconnected queues.• Each service center consists of some number of servers c working inEach service center consists of some number of servers, c, working in
parallel, upon getting to the head of the line, a customer takes the 1st
available server.
Dr. Mesut GüneşChapter 8. Queueing Models 13
Computer Science, Informatik 4 Communication and Distributed Systems
Service Times and Service MechanismService Times and Service Mechanism
Example: consider a discount warehouse where customers may:Example: consider a discount warehouse where customers may:• Serve themselves before paying at the cashier
Dr. Mesut GüneşChapter 8. Queueing Models 14
Computer Science, Informatik 4 Communication and Distributed Systems
Service Times and Service MechanismService Times and Service Mechanism
• Wait for one of the three clerks:
• Batch service (a server serving several customers simultaneously), or customer requires several servers simultaneously.
Dr. Mesut GüneşChapter 8. Queueing Models 15
Computer Science, Informatik 4 Communication and Distributed Systems
Service Times and Service MechanismService Times and Service Mechanism
Dr. Mesut GüneşChapter 8. Queueing Models 16
Computer Science, Informatik 4 Communication and Distributed Systems
ExampleExample
Candy production lineCandy production line• Three machines separated by buffers• Buffers have capacity of 1000 candies
Assumption:Allways sufficient supply of
Dr. Mesut GüneşChapter 8. Queueing Models 17
pp yraw material.
Computer Science, Informatik 4 Communication and Distributed Systems
Queueing Notation – Kendall NotationQueueing Notation Kendall Notation
Dr. Mesut GüneşChapter 8. Queueing Models 18
Computer Science, Informatik 4 Communication and Distributed Systems
Queueing Notation – Kendall NotationQueueing Notation – Kendall Notation
A notation system for parallel server queues: A/B/c/N/Ky p q• A represents the interarrival-time distribution• B represents the service-time distribution• c represents the number of parallel servers• N represents the system capacity• K represents the size of the calling population• N, K are usually dropped, if they are infinity
Common symbols for A and B• M Markov, exponential distribution• D Constant, deterministic
E E l di t ib ti f d k• Ek Erlang distribution of order k• H Hyperexponential distribution• G General, arbitrary
ExamplesExamples• M/M/1/∞/∞ same as M/M/1: Single-server with unlimited capacity and call-
population. Interarrival and service times are exponentially distributed• G/G/1/5/5: Single-server with capacity 5 and call-population 5.
Dr. Mesut GüneşChapter 8. Queueing Models 19
G/G/1/5/5: Single server with capacity 5 and call population 5.
Computer Science, Informatik 4 Communication and Distributed Systems
Queueing NotationQueueing Notation
General performance measures of queueing systems:• Pn steady-state probability of having n customers in system• Pn(t) probability of n customers in system at time t• λ arrival rate• λe effective arrival rate• μ service rate of one server• ρ server utilization
A i t i l ti b t t 1 d• An interarrival time between customers n-1 and n• Sn service time of the n-th arriving customer• Wn total time spent in system by the n-th arriving customer
W Q total time spent in the waiting line by customer• WnQ total time spent in the waiting line by customer n
• L(t) the number of customers in system at time t• LQ(t) the number of customers in queue at time t• L long run time average number of customers in system• L long-run time-average number of customers in system• LQ long-run time-average number of customers in queue• w long-run average time spent in system per customer• wQ long-run average time spent in queue per customer
Dr. Mesut GüneşChapter 8. Queueing Models 20
wQ long run average time spent in queue per customer
Computer Science, Informatik 4 Communication and Distributed Systems
Long-run Measures of Performance of Queueing SystemsLong run Measures of Performance of Queueing Systems
Dr. Mesut GüneşChapter 8. Queueing Models 21
Computer Science, Informatik 4 Communication and Distributed Systems
Long-run Measures of Performance of Queueing SystemsLong-run Measures of Performance of Queueing Systems
Primary long-run measures of performance arePrimary long run measures of performance are• L long-run time-average number of customers in system• LQ long-run time-average number of customers in queue• w long-run average time spent in system per customer• w long-run average time spent in system per customer• wQ long-run average time spent in queue per customer• ρ server utilization
Other measures of interest are• Long-run proportion of customers who are delayed longer than t0
time unitstime units• Long-run proportion of customers turned away because of
capacity constraints• Long-run proportion of time the waiting line contains more than
k0 customers
Dr. Mesut GüneşChapter 8. Queueing Models 22
Computer Science, Informatik 4 Communication and Distributed Systems
Long-run Measures of Performance of Queueing SystemsLong-run Measures of Performance of Queueing Systems
Goal of this sectionGoal of this section• Major measures of performance for a general G/G/c/N/K
queueing system• How these measures can be estimated from simulation runs
Two types of estimatorsTwo types of estimators• Sample average• Time-integrated sample averageg p g
Dr. Mesut GüneşChapter 8. Queueing Models 23
Computer Science, Informatik 4 Communication and Distributed Systems
Time-Average Number in System LTime-Average Number in System L
Consider a queueing system over a period of time TConsider a queueing system over a period of time T• Let Ti denote the total time during [0,T ] in which the system contained
exactly i customers, the time-weighted-average number in a system is defined by:defined by:
∑∑∞
=
∞
=
⎟⎠
⎞⎜⎝
⎛==00
1ˆi
i
ii T
TiiT
TL
• Consider the total area under the function is L(t), then,
∫∑∞ T11ˆ ∫∑ ===
T
ii dttL
TiT
TL
00
)(11ˆ
• The long-run time-average number of customers in system, with probability 1:
)(1ˆ LdttLLT
⎯⎯ →⎯= ∫Dr. Mesut GüneşChapter 8. Queueing Models 24
)(0
LdttLT
L T ⎯⎯ →⎯= ∞→∫
Computer Science, Informatik 4 Communication and Distributed Systems
Time-Average Number in System LTime-Average Number in System L
Number of customers in the
systemy
Time
Dr. Mesut GüneşChapter 8. Queueing Models 25
Computer Science, Informatik 4 Communication and Distributed Systems
Time-Average Number in System LTime-Average Number in System L
The time-weighted-average number in queue is:The time weighted average number in queue is:
QT
T
Qi
QiQ LdttL
TiT
TL ⎯⎯ →⎯== ∞→
∞
∫∑ 00
)(11ˆ
• G/G/1/N/K example: consider the results from the queueing system (N ≥ 4, K ≥ 3).
i TT =0
customers15120/2320/)]1(3)4(2)12(1)3(0[ˆ
==+++=L
⎩⎨⎧
≥−=
=1if1)(0 if ,0
)(L(t)tLL(t)
tLQ
customers15.120/23 ==
⎩ ≥ 1if ,1)( L(t)tL
customers 3.020
)1(2)4(1)15(0ˆ =++
=QL
Dr. Mesut GüneşChapter 8. Queueing Models 26
20
Computer Science, Informatik 4 Communication and Distributed Systems
Average Time Spent in System Per Customer w
The average time spent in system per customer, called the average
Average Time Spent in System Per Customer w
The average time spent in system per customer, called the average system time, is:
∑=N
iWw 1ˆ
where W1, W2, …, WN are the individual times that each of the N
∑=i
iWN
w1
customers spend in the system during [0,T].• For stable systems: ∞→→ Nww as ˆ
• If the system under consideration is the queue alone:
N1 ∑=
∞→⎯⎯ →⎯=N
iQN
QiQ wW
Nw
1
1ˆ
Dr. Mesut GüneşChapter 8. Queueing Models 27
Computer Science, Informatik 4 Communication and Distributed Systems
Average Time Spent in System Per Customer wAverage Time Spent in System Per Customer w
G/G/1/N/K example (cont.): p ( )The average system time is
ii64)1620(...)38(2...ˆ 521 −++−++++ WWW
The average queuing time is
unitstime6.45
)1620(...)38(25
...ˆ 521 =+++
=+++
=WWW
w
03300 ++++ unitstime2.15
03300ˆ =++++
=Qw
Dr. Mesut GüneşChapter 8. Queueing Models 28
Computer Science, Informatik 4 Communication and Distributed Systems
The Conservation Equation – Little’s LawThe Conservation Equation – Little s Law
Conservation equation (a.k.a. Little’s law)Conservation equation (a.k.a. Little s law)
wL ˆˆˆ λ= Average System time
Average # in t
Arrival rate
System timesystem
• Holds for almost all queueing systems or subsystems (regardless of the
∞→∞→= NTwL and as λ• Holds for almost all queueing systems or subsystems (regardless of the
number of servers, the queue discipline, or other special circumstances).• G/G/1/N/K example (cont.): On average, one arrival every 4 time units
and each arrival spends 4 6 time units in the system Hence at anand each arrival spends 4.6 time units in the system. Hence, at an arbitrary point in time, there are (1/4)(4.6) = 1.15 customers present on average.
Dr. Mesut GüneşChapter 8. Queueing Models 29
Computer Science, Informatik 4 Communication and Distributed Systems
Server UtilizationServer Utilization
Definition: the proportion of time that a server is busy.Definition: the proportion of time that a server is busy.• Observed server utilization, , is defined over a specified time interval
[0,T ].• Long run server utilization is ρ
ρ̂
• Long-run server utilization is ρ.• For systems with long-run stability: ∞→→ T as ˆ ρρ
Dr. Mesut GüneşChapter 8. Queueing Models 30
Computer Science, Informatik 4 Communication and Distributed Systems
Server UtilizationServer Utilization
For G/G/1/∞/∞ queues:For G/G/1/∞/∞ queues:• Any single-server queueing system with average arrival rate λ
customers per time unit, where average service time E(S) = 1/μti it i fi it it d lli l titime units, infinite queue capacity and calling population.
• Conservation equation, L = λw, can be applied.• For a stable system, the average arrival rate to the server, λ ,For a stable system, the average arrival rate to the server, λs,
must be identical to λ.• The average number of customers in the server is:
( )T
TTdttLtLT
LT
Qs0
0)()(1ˆ −
=−= ∫ TT 0∫
Dr. Mesut GüneşChapter 8. Queueing Models 31
Computer Science, Informatik 4 Communication and Distributed Systems
Server UtilizationServer Utilization
• In general, for a single-server queue:In general, for a single server queue:
ρρ =⎯⎯ →⎯= ∞→ ˆˆ LL sTs
μλλρ =⋅= )( and sE
- For a single-server stable queue: 1<=μλρ
- For an unstable queue (λ > μ), long-run server utilization is 1.
μ
Dr. Mesut GüneşChapter 8. Queueing Models 32
Computer Science, Informatik 4 Communication and Distributed Systems
Server UtilizationServer Utilization
For G/G/c/∞/∞ queues:For G/G/c/∞/∞ queues:• A system with c identical servers in parallel.• If an arriving customer finds more than one server idle, the
customer chooses a server without favoring any particular server.• For systems in statistical equilibrium, the average number of busy
servers, L , is:servers, Ls, is:
• Clearly 0 ≤ LS ≤ c μλλ == )(SELS
• The long-run average server utilization is:
λLsystems stablefor where, μλ
μλρ ccc
Ls <==
Dr. Mesut GüneşChapter 8. Queueing Models 33
Computer Science, Informatik 4 Communication and Distributed Systems
Server Utilization and System PerformanceServer Utilization and System Performance
System performance varies widely for a given utilization ρ.y p y g ρ• For example, a D/D/1 queue where E(A) = 1/λ and E(S) = 1/μ,
where:L λ/ E(S) 1/ L W 0L = ρ = λ/μ, w = E(S) = 1/μ, LQ = WQ = 0
- By varying λ and μ, server utilization can assume any value between 0 and 1.
- Yet there is never any line.• In general, variability of interarrival and service times causes
lines to fluctuate in length.lines to fluctuate in length.
Dr. Mesut GüneşChapter 8. Queueing Models 34
Computer Science, Informatik 4 Communication and Distributed Systems
Server Utilization and System PerformanceServer Utilization and System Performance
Example: A physician who • Consider the system is simulated p p yschedules patients every 10 minutes and spends Si minutes with the i-th patient:
ywith service times: S1= 9, S2=12, S3 = 9, S4 = 9, S5 = 9, ….
• The system becomes:y
⎩⎨⎧
=1.0y probabilit with minutes 129.0y probabilit with minutes 9
iS
• Arrivals are deterministic, A1 = A2 = … = λ-1 = 10.S i t h ti• Services are stochastic
- E(Si) = 9.3 min- V(S0) = 0.81 min2
- σ = 0.9 min• On average, the physician's
utilization = ρ = λ/μ = 0.93 < 1.• The occurrence of a relatively
long service time (S2 = 12) causes a waiting line to form
Dr. Mesut GüneşChapter 8. Queueing Models 35
causes a waiting line to form temporarily.
Computer Science, Informatik 4 Communication and Distributed Systems
Costs in Queueing ProblemsCosts in Queueing Problems
Costs can be associated with various aspects of the waiting line or p gservers:• System incurs a cost for each customer in the queue, say at a rate of $10
per hour per customer.W Q is the time
p p- The average cost per customer is:
Q
N Qj w
Wˆ10$
10$⋅=
⋅∑
WjQ is the time
customer j spends in queue
- If customers per hour arrive (on average), the average cost per hour is:
Qj
wN
10$1
∑=
λ̂hour is:
hour
ˆ10$ˆˆ10$
customerˆ10$
hourcustomerˆ Q
QQ L
ww ⋅
=⋅⋅=⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅⎟⎠⎞
⎜⎝⎛ λλ
• Server may also impose costs on the system, if a group of c parallel servers (1 ≤ c ≤ ∞) have utilization r, each server imposes a cost of $5 per hour while busy.
⎠⎝
Dr. Mesut GüneşChapter 8. Queueing Models 36
y- The total server cost is: ρ⋅⋅c5$
Computer Science, Informatik 4 Communication and Distributed Systems
Steady-state Behavior of Infinite-Population Markovian ModelsSteady state Behavior of Infinite Population Markovian Models
Dr. Mesut GüneşChapter 8. Queueing Models 37
Computer Science, Informatik 4 Communication and Distributed Systems
Steady-State Behavior of Markovian ModelsSteady-State Behavior of Markovian Models
Markovian models:Markovian models: • Exponential-distributed arrival process (mean arrival rate = 1/λ).• Service times may be exponentially (M) or arbitrary (G) distributed.
Q di i li i FIFO• Queue discipline is FIFO.• A queueing system is in statistical equilibrium if the probability that the
system is in a given state is not time dependent:
nn PtPntLP === )())((
• Mathematical models in this chapter can be used to obtain approximate results even when the model assumptions do not strictly hold, as a rough guide.
• Simulation can be used for more refined analysis, more faithful
Dr. Mesut GüneşChapter 8. Queueing Models 38
representation for complex systems.
Computer Science, Informatik 4 Communication and Distributed Systems
Steady-State Behavior of Markovian ModelsSteady-State Behavior of Markovian Models
Properties of processes with statistical equilibriumProperties of processes with statistical equilibrium• The state of statistical equilibrium is reached from any starting
state.• The process remain in statistical equilibrium once it has reached
it.
Dr. Mesut GüneşChapter 8. Queueing Models 39
Computer Science, Informatik 4 Communication and Distributed Systems
Steady-State Behavior of Markovian ModelsSteady-State Behavior of Markovian Models
For the simple model studied in this chapter, the steady-stateFor the simple model studied in this chapter, the steady state parameter, L, the time-average number of customers in the system is:
∞
• Apply Little’s equation L=λ w to the whole system and to the
∑=
=0n
nnPL
• Apply Little s equation, L=λ w, to the whole system and to the queue alone:
QQQ wLwwLw λλ
=−== ,1 ,
G/G/c/∞/∞ example: to have a statistical equilibrium, a d ffi i t diti i
QQQ μλ
necessary and sufficient condition is:
1<=λρ
Dr. Mesut GüneşChapter 8. Queueing Models 40
μρ
c
Computer Science, Informatik 4 Communication and Distributed Systems
M/G/1 QueuesM/G/1 Queues
Single-server queues with Poisson arrivals and unlimited capacity.S g e se e queues o sso a a s a d u ed capac ySuppose service times have mean 1/μ and variance σ 2 and ρ = λ / μ < 1, the steady-state parameters of M/G/1 queue:
λ
1
0 ρμλρ
−=
=
P
)1(2
)1(
1222
0
ρμσρρ
ρ
−+
+=L
PThe particular
distribution is not known!
)1(2)/1(1 22
ρσμλ
μ −+
+=w
)/1()1(2
)1(
22
222
λρ
μσρ−
+=QL
Dr. Mesut GüneşChapter 8. Queueing Models 41
)1(2)/1( 22
ρσμλ
−+
=Qw
Computer Science, Informatik 4 Communication and Distributed Systems
M/G/1 QueuesM/G/1 Queues
There are no simple expressions for the steady-state probabilities P0,There are no simple expressions for the steady state probabilities P0, P1, P2 ,…L – LQ = ρ is the time-average number of customers being served.Average length of queue, LQ, can be rewritten as:
)1(2)1(2
222
ρσλ
ρρ
−+
−=QL
• If λ and μ are held constant, LQ depends on the variability, σ 2, of the service times.
Dr. Mesut GüneşChapter 8. Queueing Models 42
Computer Science, Informatik 4 Communication and Distributed Systems
M/G/1 QueuesM/G/1 Queues
Example: Two workers competing for a job, Able claims to be faster than B k b B k l i b iBaker on average, but Baker claims to be more consistent,
• Poisson arrivals at rate λ = 2 per hour (1/30 per minute).• Able: 1/μ = 24 minutes and σ 2 = 202 = 400 minutes2:
customers 711.2)5/41(2
]40024[)30/1( 22=
−+
=QL
- The proportion of arrivals who find Able idle and thus experience no delay isP0 = 1-ρ = 1/5 = 20%.
• Baker: 1/μ = 25 minutes and σ 2 = 22 = 4 minutes2:Baker: 1/μ 25 minutes and σ 2 4 minutes :
customers 097.2)6/51(2
]425[)30/1( 22=
−+
=QL
- The proportion of arrivals who find Baker idle and thus experience no delay is P0 = 1-ρ = 1/6 = 16.7%.
• Although working faster on average Able’s greater service variability results in an
Dr. Mesut GüneşChapter 8. Queueing Models 43
Although working faster on average, Able s greater service variability results in an average queue length about 30% greater than Baker’s.
Computer Science, Informatik 4 Communication and Distributed Systems
M/M/1 QueuesM/M/1 Queues
Suppose the service times in an M/G/1 queue are exponentiallySuppose the service times in an M/G/1 queue are exponentially distributed with mean 1/μ, then the variance is σ 2 = 1/μ2. • M/M/1 queue is a useful approximate model when service times
h t d d d i ti i t l l t th ihave standard deviation approximately equal to their means.• The steady-state parameters
λ
( )1ρλ
ρρμ
ρ
−=
=
nnP ρ−= 10P
)1(11
1
λ
ρρ
λμλ
==
−=
−=
w
L
( ) 1
)1(22
ρρ
λμμλ
ρμλμ
−=
−=
−−
QL
Dr. Mesut GüneşChapter 8. Queueing Models 44
( ) )1( ρμρ
λμμλ
−=
−=Qw
Computer Science, Informatik 4 Communication and Distributed Systems
M/M/1 QueuesM/M/1 Queues
Single-chair unisex hair-styling shopSingle chair unisex hair styling shop• Interarrival and service times are exponentially distributed• λ = 2 customers/hour and µ = 3 customers/hour
==32
μλρ Customers 2
232
=−
=−
=λμ
λL
⎟⎞
⎜⎛
=−=
1
0
221311
P
P ρ
h2111
hour 122
===λLw
=⎟⎠⎞
⎜⎝⎛⋅=
=⎟⎠⎞
⎜⎝⎛⋅=
2
2
1
421
933
P
P
Customers34
)23(34
)(
hour33
1
2
===
=−=−=
λλμ
Q
Q
L
ww
∑=
≥ =−=
⎟⎠
⎜⎝
3
04
2
81161
2733
nnPP Customers 2
32
34
3)23(3)(
=+=+=
−−
μλ
λμμ
Q
Q
LL
Dr. Mesut GüneşChapter 8. Queueing Models 45
0n
Computer Science, Informatik 4 Communication and Distributed Systems
M/M/1 QueuesM/M/1 Queues
Example: M/M/1 queue with p qservice rate μ =10 customers per hour.• Consider how L and w
λ 5 6 7.2 8.64 10ρ 0.5 0.60 0.72 0.864 1• Consider how L and w
increase as arrival rate, λ, increases from 5 to 8.64 by increments of 20%
L 1.0 1.50 2.57 6.350 ∞w 0.2 0.25 0.36 0.730 ∞
increments of 20%
• If λ/μ ≥ 1, waiting lines tend to continually grow in length 14
16
18
20Lw
continually grow in length
• Increase in average system 8
10
12
14
mbe
r of C
usto
mer
stime (w) and average number in system (L) is highly nonlinear as a function of ρ.
0
2
4
6Num
Dr. Mesut GüneşChapter 8. Queueing Models 46
0 0.5 0.6 0.7 0.8 0.9 1
rho
Computer Science, Informatik 4 Communication and Distributed Systems
Effect of Utilization and Service VariabilityEffect of Utilization and Service Variability
For almost all queues, if lines are too long, they can be reduced byFor almost all queues, if lines are too long, they can be reduced by decreasing server utilization (ρ) or by decreasing the service time variability (σ2).A f th i bilit f di t ib tiA measure of the variability of a distribution,• coefficient of variation (cv):
)(XV[ ]2
2
)()()(
XEXVcv =
• The larger cv is, the more variable is the distribution relative to its expected value
• For exponential service times with rate µ- E(X) = 1/µ- V(X) = 1/µ2
Dr. Mesut GüneşChapter 8. Queueing Models 47
( ) µcv = 1
Computer Science, Informatik 4 Communication and Distributed Systems
Effect of Utilization and Service VariabilityEffect of Utilization and Service Variability
Consider LQ for any M/G/1 queue:Q y q
+=
)1( 222L μσρ
⎟⎟⎞
⎜⎜⎛ +
⎟⎟⎞
⎜⎜⎛
=
−=
)(1
)1(222 cv
LQ
ρ
ρ
⎟⎠
⎜⎝
⎟⎠
⎜⎝ − 21 ρ
L for M/M/1LQ for M/M/1queue
C t th M/M/1Corrects the M/M/1formula to account
for a non-exponential service time dist’n
Dr. Mesut GüneşChapter 8. Queueing Models 48
Computer Science, Informatik 4 Communication and Distributed Systems
Multiserver Queue – M/M/cMultiserver Queue – M/M/c
M/M/c/∞/∞ queue: c servers operating in parallelM/M/c/∞/∞ queue: c servers operating in parallel• Arrival process is poisson with rate λ• Each server has an independent and identical exponential service-time
distribution with mean 1/distribution, with mean 1/μ.• To achieve statistical equilibrium, the offered load (λ/μ) must satisfy
λ/μ < c, where λ/(cm) = ρ is the server utilization.
1
Waiting line
Calling population λ
2
Dr. Mesut GüneşChapter 8. Queueing Models 49
c
Computer Science, Informatik 4 Communication and Distributed Systems
Multiserver Queue – M/M/cMultiserver Queue – M/M/c
• The steady-state parameters for M/M/cThe steady state parameters for M/M/c
μλρc
=
λμμ
μλμλ
cc
cnP
cc
n
n
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎥
⎦
⎤⎢⎣
⎡=
−−
=∑ !
1!
)/(1
1
00
( )
( )ρρρ
ρ
cLPPcc
PccLP
c
c
≥∞⋅
−=≥∞
+ )()()1(!
)()(
1
0
Probability that ll ( )
λ
ρρρ
ρρρ
Lw
cLPccc
PccL
=
−≥∞
+=−
+=1
)()1)(!(
)(2
0all servers are busy
( )ρ
ρλ
cLPLQ −≥∞⋅
=1
)(
Dr. Mesut GüneşChapter 8. Queueing Models 50
ρcLL Q =−
Computer Science, Informatik 4 Communication and Distributed Systems
Multiserver Queue – M/M/cMultiserver Queue – M/M/c
Probability of empty system Number of customers in systemProbability of empty system Number of customers in system
Dr. Mesut GüneşChapter 8. Queueing Models 51
Computer Science, Informatik 4 Communication and Distributed Systems
Multiserver Queue – Common ModelsMultiserver Queue – Common Models
Other common multiserver queueing modelsOther common multiserver queueing models
⎟⎞
⎜⎛ +⎟
⎞⎜⎛ )(1 22 cvρ
⎟⎟⎠
⎞⎜⎜⎝
⎛ +⎟⎟⎠
⎞⎜⎜⎝
⎛−
=2
)(11
cvLQ ρρ
LQ for M/M/1queue
Corrects the M/M/1formula
• M/G/c/∞: general service times and c parallel server. The parameters can be approximated from those of the M/M/c/∞/∞ model.
f f• M/G/∞: general service times and infinite number of servers.• M/M/c/N/∞: service times are exponentially distributed at rate μ and c
servers where the total system capacity is N ≥ c customer. When an arrival
Dr. Mesut GüneşChapter 8. Queueing Models 52
occurs and the system is full, that arrival is turned away.
Computer Science, Informatik 4 Communication and Distributed Systems
Multiserver Queue – M/G/∞Multiserver Queue – M/G/∞
• M/G/∞: general service times and infinite number of serversg- customer is its own server- service capacity far exceeds service demand- when we want to know how many servers are required so thatwhen we want to know how many servers are required so that
customers are rarely delayed
( )10==
−n
neP μλ μ
λ
,1,0,!
0 =
==
−
n
eP
nn
eP
μλ
K
0
1=w
μ0
=
=Q
L
w
μλ
Dr. Mesut GüneşChapter 8. Queueing Models 53
0=QLμ
Computer Science, Informatik 4 Communication and Distributed Systems
Multiserver Queue – M/G/∞Multiserver Queue – M/G/∞
How many users can be logged in simultaneously in a computer y gg y psystem• Customers log on with rate λ = 500 per hour• Stay connected in average for 1/µ =180 minutes = 3 hoursy g µ• For planning purposes it is pretended that the simultaneous logged in
users is infinite• Expected number of simultaneous users Lp
15003500 =⋅==μλL
• To ensure providing adequate capacity 95% of the time, the number of parallel users c has to be restricted
μ
parallel users c has to be restricted
∑∑=
−
=
≥==≤∞c
n
nc
nn n
ePcLP0
1500
095.0
!)1500())((
Dr. Mesut GüneşChapter 8. Queueing Models 54
• The capacity c =1564 simultaneous users satisfies this requirement
Computer Science, Informatik 4 Communication and Distributed Systems
Multiserver Queue with Limited CapacityMultiserver Queue with Limited Capacity
M/M/c/N/∞: service times are exponentially distributed at rate μ and p y μc servers where the total system capacity is N ≥ c customer• When an arrival occurs and the system is full, that arrival is turned away• Effective arrival rate λ is defined as the mean number of arrivals per• Effective arrival rate λe is defined as the mean number of arrivals per
time unit who enter and remain in the systemaaP
c Ncn
cn
ρ ⎥⎦
⎤⎢⎣
⎡++=
−
−∑ ∑!!1
1
0
λa =
PccaP
cn
cN
N
N
n cn
ρ
=
⎥⎦
⎢⎣
−
= +=∑ ∑
!
!!
0
1 10
μλρ
μ
c=
( )P
cNc
aPL
Ne
cNcNc
Q
λλ
ρρρρρ
−=
−−−−−
= −−
)1(
)1()(1)1(!
0
μ
ww
Lw
e
QQ λ
+=
=
1
(1 - PN) probability that a customer will find a space and be able to
Dr. Mesut GüneşChapter 8. Queueing Models 55
wL
ww
e
Q
λμ
=
+= space and be able to enter the system
Computer Science, Informatik 4 Communication and Distributed Systems
Multiserver Queue with Limited CapacityMultiserver Queue with Limited Capacity
Single-chair unisex hair-styling shop (again!)• Space only for 3 customers:
one in service and two waiting• First computer P0
• Queue time
246.011428
Lw
λ
• System time, time in shop415.0
32
32
321
13
2
10 =
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛++
=
∑=
−
n
nP
114eλ
661• P(system is full)
• Expected number of( )
123.0658
1!1 02
332
3 ==== PPPN
579.0114661
==+=μQww
• Average of the queue• Expected number of
customers in shop651!1
431.0=QL015166
=== wL λ
• Effective arrival rate• Probability of busy shop
754.1114812 ==⎟⎞
⎜⎛ −=λ
015.165
wL eλ
Dr. Mesut GüneşChapter 8. Queueing Models 56
754.16565
12 ⎟⎠
⎜⎝
eλ
585.01 0 ==−μλeP
Computer Science, Informatik 4 Communication and Distributed Systems
Steady-state Behavior of Finite-Population ModelsSteady state Behavior of Finite Population Models
Dr. Mesut GüneşChapter 8. Queueing Models 57
Computer Science, Informatik 4 Communication and Distributed Systems
Steady-State Behavior of Finite-Population ModelsSteady-State Behavior of Finite-Population Models
In practical problems calling population is finite• When the calling population is small, the presence of one or more customers in
the system has a strong effect on the distribution of future arrivals.Consider a finite-calling population model with K customers (M/M/c/K/K)
Th ti b t th d f i i it d th t ll f i i• The time between the end of one service visit and the next call for service is exponentially distributed with mean = 1/λ.
• Service times are also exponentially distributed with mean 1/µ.• c parallel servers and system capacity is Kc parallel servers and system capacity is K.
K C1
Waiting line with capacity K
2
K CustomersMean runtime
1/λ
2
Dr. Mesut GüneşChapter 8. Queueing Models 58
c
Computer Science, Informatik 4 Communication and Distributed Systems
Steady-State Behavior of Finite-Population ModelsSteady-State Behavior of Finite-Population Models
• Some of the steady-state probabilities of M/M/c/K/K :So e o t e steady state p obab t es o / /c/ /
μλ
μλ
ccnKK
nK
PK n
cn
c n
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
+⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛= ∑∑
−
−
−
!)!(!
11
00
μλ
μμ
cnPnK
P
ccnKnn
cnn
⎪⎪
⎨
⎧−=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
⎥⎦⎢⎣ ⎠⎝⎠⎝⎠⎝ ==
1,...,1,0 ,
!)!(
0
0
μλ
μ
KccnccnK
KP n
cn
n
⎪⎪
⎩
⎪⎨
+=⎟⎟⎠
⎞⎜⎜⎝
⎛−
⎠⎝⎠⎝=
− ,...1, ,!)!(
!
μλρλc
LwnPL ee
K
nn === ∑
=
,/ ,0
∑ −=K
ne PnK )( λλ
service)xitingentering/e(or queuetocustomersofratearrivaleffectiverun long theiswhere eλ
Dr. Mesut GüneşChapter 8. Queueing Models 59
∑=n
ne0
)(
Computer Science, Informatik 4 Communication and Distributed Systems
Steady-State Behavior of Finite-Population ModelsSteady-State Behavior of Finite-Population Models
Example: two workers who are responsible for 10 milling p p gmachines. • Machines run on the average for 20 minutes, then require an
average 5-minute service period both times exponentiallyaverage 5 minute service period, both times exponentially distributed: λ = 1/20 and μ = 1/5.
• All of the performance measures depend on P0:1
⎤⎡−
065.0205
2!2)!10(!10
20510
110
22
12
00 =
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
−+⎟
⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛=
=−
−
=∑∑n
n
nn
n
nnP
• Then, we can obtain the other Pn, and can compute the expected number of machines in system:
10
• The average number of running machines:
machines 17.30
== ∑=n
nnPL
Dr. Mesut GüneşChapter 8. Queueing Models 60
The average number of running machines:machines 83.617.310 =−=− LK
Computer Science, Informatik 4 Communication and Distributed Systems
Networks of QueuesNetworks of Queues
Dr. Mesut GüneşChapter 8. Queueing Models 61
Computer Science, Informatik 4 Communication and Distributed Systems
Networks of QueuesNetworks of Queues
Many systems are naturally modeled as networks of single queuesy y y g q• Customers departing from one queue may be routed to another
The following results assume a stable system with infinite callingThe following results assume a stable system with infinite calling population and no limit on system capacity:• Provided that no customers are created or destroyed in the queue, then
the departure rate out of a queue is the same as the arrival rate into thethe departure rate out of a queue is the same as the arrival rate into the queue, over the long run.
• If customers arrive to queue i at rate λi, and a fraction 0 ≤ pij ≤ 1 of them are routed to queue j upon departure, then the arrival rate from queue i
Dr. Mesut GüneşChapter 8. Queueing Models 62
q j p p , qto queue j is λi pij over the long run.
Computer Science, Informatik 4 Communication and Distributed Systems
Networks of QueuesNetworks of Queues
• The overall arrival rate into queue j:e o e a a a ate to queue j
∑+=i
ijijj pa all
λλ
Arrival rate from outside the network
Sum of arrival rates from other queues in
network
• If queue j has cj < ∞ parallel servers, each working at rate μj, then the long-run utilization of each server is: (where ρj < 1 for stable queue).
jλ
• If arrivals from outside the network form a Poisson process with rate ajfor each queue j and if there are c identical servers delivering
jj
jj c μ
ρ =
for each queue j, and if there are cj identical servers delivering exponentially distributed service times with mean 1/μj, then, in steady state, queue j behaves likes an M/M/cj queue with arrival rate
∑Dr. Mesut GüneşChapter 8. Queueing Models 63
∑+=i
ijijj pa all
λλ
Computer Science, Informatik 4 Communication and Distributed Systems
Network of QueuesNetwork of Queues
C t 80c = ∞0.4
CustCustomer Population
80 c = 1
0.6
Cust.hour
Discount store example: • Suppose customers arrive at the rate 80 per hour and
40% choose self-service.40% choose self service. • Hence:
- Arrival rate to service center 1 is λ1 = 80(0.4) = 32 per hour- Arrival rate to service center 2 is λ2 = 80(0.6) = 48 per hour.2 ( ) p
• c2 = 3 clerks and μ2 = 20 customers per hour.• The long-run utilization of the clerks is:
ρ2 = 48/(3*20) = 0.8ρ2 ( )• All customers must see the cashier at service center 3,
the overall rate to service center 3 is λ3 = λ1 + λ2 = 80per hour.
Dr. Mesut GüneşChapter 8. Queueing Models 64
- If μ3 = 90 per hour, then the utilization of the cashier is:ρ3 = 80/90 = 0.89
Computer Science, Informatik 4 Communication and Distributed Systems
SummarySummary
Introduced basic concepts of queueing models.oduced bas c co cep s o queue g ode sShowed how simulation, and some times mathematical analysis, can be used to estimate the performance measures of a system.Commonly used performance measures: L L w w ρ and λCommonly used performance measures: L, LQ, w, wQ, ρ, and λe.When simulating any system that evolves over time, analyst must decide whether to study transient behavior or steady-state behavior.
Si l f l i t f th t d t t b h i f• Simple formulas exist for the steady-state behavior of some queues.Simple models can be solved mathematically, and can be useful in providing a rough estimate of a performance measure.
Dr. Mesut GüneşChapter 8. Queueing Models 65