Chapter 6
6.2 Queueing Notation
6.3 Long-Run Measures of Performance of Queueing
Systems
6.4 Steady-State Behavior of InfinitePopulation Markovian Models
6.2 Queueing Notation
• A notation system for parallel server queues: A/B/c/N/K
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–
–
–
–
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.
– Probably the most common assumptions are that the queue is either
M/G/1 a single-server queue with Poisson arrivals and unlimited
capacity or
M/M/1 a single-server queue with Poisson arrivals, service times
exponentially distributed , and unlimited capacity
Additional Notation Used in the Text
(This is common usage of these variables)
• Primary performance measures of queueing systems:
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–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
Pn:
Pn(t):
l:
le:
m:
r:
An:
Sn:
Wn:
WnQ:
L(t):
LQ(t):
L:
LQ:
w:
wQ:
steady-state probability of having n customers in system,
probability of n customers in system at time t,
arrival rate,
effective arrival rate,
service rate of one server,
server utilization,
interarrival time between customers n-1 and n,
service time of the nth arriving customer,
total time spent in system by the nth arriving customer,
total time spent in the waiting line by customer n,
the number of customers in system at time t,
the number of customers in queue at time t,
long-run time-average number of customers in system,
long-run time-average number of customers in queue,
long-run average time spent in system per customer,
long-run average time spent in queue per customer.
6.3 Long-Run Measures of Performance of
Queueing Systems
• The systems is usually the waiting line plus the service
mechanism
– Can be a subsystem of the queueing system
• Primary long run measures are
– Long-run time-average number of customers in the system and in the
queue
• the L’s
– Long-run average time spent in the system and in the queue per
customer
• the W’s
– The server utilization or proportion of the time the server is busy
• the r
Time-Average Number of Customers in System: L
• Consider 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:
1
Lˆ 
T


i 0

iTi 
T 
i i 
T
i 0 

– Consider the total area under the function is L(t), then,
1
Lˆ 
T


1
iTi 
T
i 0
T
 L(t )dt
0
– The long-run time-average # in system, with probability 1:
1
Lˆ 
T
T
 L(t)dt  L
as T  
0
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Time-Average Number of Customers in System: L
– The time-weighted-average number in queue is:
1
LˆQ 
T


1
iTi 
T
i 0
Q
T
L
0
Q (t )dt
 LQ as T  
– G/G/1/N/K example: consider the results from the queueing system (N
> 4, K > 3).
Lˆ  [0(3)  1(12)  2(4)  3(1)] / 20
 23 / 20  1.15 cusomters
if L( t) 0
0,
LQ (t )  
 L(t )  1, if L( t) 1
0(15)  1(4)  2(1)
LˆQ 
 0.3 customers
20
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Average Time Spent in System Per Customer: w
• The average time spent in system per customer, called the
N
average system time, is:
1
wˆ 
W

N
i
i 1
where W1, W2, …, WN are the time each customer spends in the system
during [0,T] and N is the number of arrivals during [0,T].
– For stable systems:
wˆ  w as N  
– If the system under consideration is the queue alone:
1 N Q
wˆ Q  Wi  wQ as N  
N i 1
– G/G/1/N/K example (cont.): the average system time is
ˆ
w
W1  W2  ...  W5 2  (8  3)  ...  (20  16)

 4.6 time units
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5
7
The Conservation Equation
• Conservation equation (a.k.a. Little’s law)
Average # in
system
Lˆ  lˆwˆ
Average
System time
Arrival rate
L  lw as T   and N  
– 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 an
arbitrary point in time, there is (1/4)(4.6) = 1.15 customers present on
average.
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Server Utilization
• Definition: the proportion of time that a server is busy.
– Observed server utilization, rˆ , is defined over a specified time interval
[0,T].
– Long-run server utilization is r.
rˆ  r as T  
– For systems with long-run stability:
– See page 214
observed server utilization = (total busy time) / T
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Server Utilization: For G/G/1/∞/∞ queues
• single-server queueing system with average arrival rate l
customers per time unit and the average service rate of the
server is m, infinite queue capacity and calling population
– The average service time E(S) = 1/m time units
– Conservation equation, L = lw, can be applied.
– For a stable system, the average arrival rate to the server, ls, must be
identical to l.
– The average number of customers in the server is:
1
Lˆs 
T
T  T0


L
(
t
)

L
(
t
)
dt

Q
0
T
T
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Server Utilization: For G/G/1/∞/∞ queues (2)
– In general, for a single-server queue:
Lˆ s  rˆ  Ls  r as T  
and
•
r  lE ( s ) 
l
m
l
1
For a single-server stable queue:
m
– The arrival rate must be less than the service rate for the single
server queue to be stable.
r
• For an unstable queue (l > m), long-run server
utilization is 1.
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Server Utilization: For G/G/1/∞/∞ 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, Ls, is: Ls, = lE(s) = l/m.
• The long-run average server utilization is:
Ls
l
r

, where l  cm for stablesystems
c cm
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Server Utilization and System Performance P. 217
• System performance varies widely for a given utilization r.
– For example, a D/D/1 queue (D is deterministic) where E(A) = 1/l
and E(S) = 1/m, where:
L = r = l/m, w = E(S) = 1/m, LQ = WQ = 0.
• By varying l and m, server utilization can assume any
value between 0 and 1.
• Yet there is never any line: See Figure 6.10
• In general, variability of interarrival and service times causes
lines to fluctuate in length.
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Server Utilization and System Performance P. 217
• Example: A physician who schedules patients every 10 minutes and spends Si
minutes with the ith patient:
 9 minuteswith probability 0.9
Si  
12 minuteswith probability 0.1
–
–
–
–
Arrivals are deterministic, A1 = A2 = … = l-1 = 10.
Services are stochastic, E(Si) = 9.3 min and V(S0) = 0.81 min2.
On average, the physician's utilization = r  l/m = 0.93 < 1.
Consider the system is simulated with service times: S1 = 9, S2 = 12, S3 =
9, S4 = 9, S5 = 9, …. The system becomes:
– The occurrence of a relatively long service time (S2 = 12) causes a waiting
line to form temporarily.
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6.3.5 Costs in Queueing Problems
• Costs can be associated with various aspects of the waiting line or
servers:
– System incurs a cost for each customer in the queue, say at a rate of $10 per
hour per customer.
• The average cost per customer is:
N

j 1
$10*W jQ
N
 $10* wˆ Q
WjQ is the time
customer j spends in
queue
• If lˆ customers per hour arrive (on average), the average cost
per hour is:
$10* wˆ
 ˆ customer
Q 
  $10* lˆwˆ Q  $10* LˆQ / hour
l

hour  customer

– 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.
• The total server cost is: $5*cr.
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