queuingTheiry.pdf

Queueing Theory

Results

7.1 NOTATION FOR QUEUE TYPES

The following standard notation is used to characterize systems with a single queue in equilibrium, identical parallel servers, unlimited waiting room, and first-come, first-served queueing discipline. Each system is defined by

A/B/m

where

A denotes the distribution of interarrival times B denotes the distribution of service times m denotes the number of servers in parallel

Commonly used symbols for both the A and B positions in this notation are M and G. The symbol M is used for the exponential distribution to denote its Markovian (“memoryless”) property. The symbol G is used for a general distri-bution of independent and identically distributed random variables. For example, an M/G/1 queue has exponentially distributed interarrival times, a general dis-tribution for service times, and one server. A G/G/m queue has general distri-butions for interarrival times and service times, and m servers in parallel.

(Gross and Harris, 1985; Kendall, 1953; Wolff, 1989).

7.2 DEFINITIONS OF QUEUEING SYSTEM VARIABLES

Lq = average queue length (average number of customers in queue)

L = average system length (average number of customers in system, including those being served)

Wq = average waiting time in queue (average time a customer spends in queue)

W = average time in system (average time a customer spends in queue plus service)

N = number of customers in system (E[N] = L) T = time customer spends in system (E[T] = W) m = number of servers

λ = arrival rate (number of customers arriving per unit time); 1/λ = mean interarrival time

µ = service rate at one server (number of customers served per unit time); 1/µ = mean service time

ρ = = Traffic intensity (ρ < 1) σa2 = variance of interarrival times σs2 = variance of service times

C_a2 ₌ λ2σ

a2= Squared coefficient of variation of interarrival times

Cs2 = µ2σs2= Squared coefficient of variation of service times.

7.3 LITTLE’S LAW AND GENERAL QUEUEING SYSTEM

RELATIONSHIPS

Lq = λWq (Little’s Law) (7.1)

L = λW (Little’s Law) (7.2)

(7.3)

(7.4)

λ mµ

L=Lq+

µ λ

(Allen, 1978; Gillett, 1976; Gross and Harris, 1985; Hillier and Lieberman, 1980; Hopp and Spearman, 1996; Little, 1961; Medhi, 1991). Note: Equations 7.1–7.4 hold for all queue types defined in Section 7.1.

7.4 EXTENSION OF LITTLE’S LAW

For the M/G/1 queue, Little’s Law L = λW can be extended to higher moments. For the k th moment:

E[N(N – 1)(N – 2)…(N – k + 1)] = λk_E[Tk_{] (7.5)} where

N = Number of customers in system T = Time customer spends in system Special cases:

k = 1: E[N] = λE[T] (i.e., L = λW) (7.6) k = 2: E[N(N – 1)] = λ2 _E[T2_{] (7.7)} Hence

7.5 FORMULAS FOR AVERAGE QUEUE LENGTH

L

where

m = number of servers

ρ = = = Traffic intensity (ρ < 1)

C_a2 ₌ λ2σ

a2 = (Arrival rate)2× (Variance of interarrival times)

C_s2 ₌ µ2σ

s2 = (Service rate of one server)2× (Variance of service times) References for L_q and W_q results are given in the table in Section 7.7. For the case m = 1, Equations 7.9 and 7.10 are exact, Equation 7.11 is approximate. For general m, Equation 7.12 is exact, Equations 7.13, 7.14, and 7.15 are approximate. The approximations for M/G/m and G/G/m with general m (Equations 7.14 and 7.15) can be improved slightly by replacing the first term, , by the more complicated but exact expression given in Equation 7.12.

Queue

Type m = 1 General m

M/M/m (7.9)

(7.12)

(7.13)

M/G/m (7.10) _. (7.14)

G/G/m . (7.11) (7.15) ρ ρ 2 1−           m m m m m k m m k k m m ρ ρ ρ _ρ ρ ρ ρ ρ

( )

(

) ( )

(

) ( )

∑

Cs ρ

ρ

2 1 2

1 1 2 m s C + ( ) − +     ρ ρ

2 2 2

1− 2

+  

 

Ca Cs ρ

ρ

2 1 2 2

1 2 m a s C C + ( ) − +     λ µ m Arrival rate Total service rate

ρ 2 1 ρ 1

m+

7.6 FORMULAS FOR AVERAGE TIME IN QUEUE

W

where

m = number of servers

ρ = = = Traffic intensity (ρ < 1)

C_a2 ₌ λ2σ a

2 _{= (Arrival rate)}2× _{(Variance of interarrival times)} C_s2 ₌ µ2σ

s2 = (Service rate of one server)2× (Variance of service times) References for L_q and W_q results are given in the table in Section 7.7. For the case m = 1, Equations 7.16 and 7.17 are exact, Equation 7.18 is approximate. For general m, Equation 7.19 is exact, Equations 7.20, 7.21, and 7.22 are approximate. The approximations for M/G/m and G/G/m with general m (Equations 7.21 and 7.22) can be improved slightly by replacing the first term, , by the more complicated but exact expression given in Equation 7.19.

Queue

Type m = 1 General m

M/M/m (7.16)

M/G/m (7.17) (7.21)

G/G/m (7.18) (7.22)

1 1 µ −     ρ ρ           m mm m m m k m m m k k m m ρ ρ ρ ρ ρ ρ ρ

( )

µ

(

) ( )

+ −

(

) ( )

µ

(

)

= − + ( )−

∑

2 1 1

approx. 1 1 1 2 2 µ −     +     ρ ρ

Cs ρ

ρ

2 1 1 2

1 1 2 m s m C + ( )− µ

(

)

+     1 1 2 2 2 µ −     +     ρ ρ

Ca Cs ρ

ρ

2 1 1 2 2

1 2 m a s m C C + ( )− µ

(

)

+     λ µ m Arrival rate Total service rate

ρ 2 1 1 ρ

1 m

m

+

7.7 REFERENCES FOR FORMULAS FOR AVERAGE QUEUE

LENGTH AND TIME IN QUEUE

(given in Equations (7.9)–(7.15) and Equations (7.16)–(7.22))

Queue

Type m = 1 General m

M/M/m Allen, 1978 For exact Equations 7.12 and 7.19: Cohen, 1985 Allen, 1978

Gillett, 1976 Buzacott and Shanthikumar, 1993 Gross and Harris, 1985 Gillett, 1976

Hopp and Spearman, 1996 Gross and Harris, 1985 Morse, 1958 Hall, 1991

Wolff, 1989 Hillier and Lieberman, 1980

For approximate Equations 7.13 and 7.20: Hopp and Spearman, 1996

Sakasegawa, 1977 Whitt, 1993

M/G/m Cox and Smith, 1971 Gross and Harris, 1985 Hall, 1991

Kendall, 1951 Kleinrock, 1975

Sakasegawa, 1977 Sakasegawa, 1977

G/G/m Hall, 1991 Hall, 1991

Hopp and Spearman, 1996 Hopp and Spearman, 1996 Sakasegawa, 1977 Sakasegawa, 1977 Tanner, 1995 Tanner, 1995 Shanthikumar and

Buzacott, 1980

7.8 POLLACZEK-KHINTCHINE FORMULA FOR AVERAGE

TIME IN QUEUE

W

Equation 7.17 for the average time in queue W_q in an M/G/1 queue is sometimes called the Pollaczek-Khintchine formula. It is an exact result, based on derivations by Pollaczek (1930) and Khintchine (1932). More recent references are given in Section 7.7.

7.9 ADDITIONAL FORMULAS FOR AVERAGE TIME IN

QUEUE

W

The average time in queue Wq in an G/G/1 queue is given by the approximate result in Equation 7.18. An earlier approximation for W_q in a G/G/1 queue, derived by Kingman for the case of heavy traffic (traffic intensity ρ close to 1) is given by

(7.23)

(Kingman, 1961, 1965; Kleinrock, 1976; Larson and Odoni, 1981; Medhi, 1991; Tanner, 1995).

In terms of coefficients of variation Ca and Cs, Kingman’s result (Equation 7.23) becomes

For ρ close to 1, this expression and Equation 7.18 are in close agreement. Numerical comparisons for the performance of these formulas are given in Tanner (1995). Note that, when interarrival times are exponentially distributed (C_a = 1), Equation 7.18 reduces to the exact result for Wq in an M/G/1 queue, given by Equation 7.17.

G G/ / :1 Wq a s

1

1 2

2 2

≅ −  

 

+  

  λ

ρ

σ σ

Wq C C

a s

≅ µ −

 





(

)



 _ 1

1 2

2 2

ρ ρ

A refinement to Equation 7.18 for Wq in a G/G/1 queue is given in Krämer and Langenbach-Belz (1976, 1978):

(7.24)

where

For analyzing networks of queues, Whitt uses this refinement in the case C2_{a< 1,} and the simpler result (Equation 7.18) in the case Ca2≥ 1 (Whitt, 1983).

7.10

HEAVY TRAFFIC APPROXIMATION FOR

DISTRIBUTION OF TIME IN QUEUE

For a G/G/1 queue, let

t be the time spent in the queue

f(t) be the probability density function of t

F(t) be the cumulative distribution function of t, where

F(t) = = Pr{time in queue ≤ t}

Under heavy traffic conditions (traffic intensity ρ = λ/µ close to 1), the distribution of time spent in a G/G/1 queue can be approximated by an exponential distribution, i.e.,

(7.25) G G/ / :1 Wq Ca Cs g ,C Ca, s

1 1 2 2 2 2 2 ≅ µ −     +    

(

)

g C C

C

C C C

C

C C C

a s a a s a a a s a ρ ρ ρ ρ , , exp exp 2 2 2 2 2 2 2 2 2 2 2 1 3 1 1 1 1 4 1

(

)

(

)

(

)

(

)



(

)

−

(

)

(

)

(

)

(

)

f x dx t

( )

∫

0

f t Wqe

and

(7.26) where Wq is the average time in queue for a G/G/1 queue (Kingman, 1961, 1962, 1965; Kleinrock, 1976; Larson and Odoni, 1981; Medhi, 1991).

This heavy traffic approximation can also be applied to a queue with multiple severs in parallel (i.e., a G/G/m queue). When traffic intensity is close to 1, the distribution of time spent in a G/G/m queue can be approximated by the exponential distribution given by Equations 7.25 and 7.26, where Wq in this case is the average time in queue for a G/G/m queue (Kingman, 1965; Kleinrock, 1976; Larson and Odoni, 1981; Medhi, 1991; Whitt, 1993).

Approximations for the average time in queue Wq for G/G/1 and G/G/m queues are given in Sections 7.6 and 7.9.

7.11

QUEUE DEPARTURE PROCESS

For a G/G/1 queue, let

ν = departure rate (number of customers departing after service per unit time); 1/ν = mean inter-departure time

σd2 = variance of inter-departure times

Cd 2

= ν2σd2 _{= squared coefficient of variation of inter-departure times} For traffic intensity ρ < 1, the departure rate must equal the arrival rate, i.e.,

ν = λ (7.27)

The squared coefficient of variation of inter-departure times Cd 2

is given approxi-mately by

(7.28)

(Hopp and Spearman, 1996; Whitt, 1983). F t

( )

ρ= λ µ m

Cd Cs Ca

2 2 2 2 2

7.12

DISTRIBUTION RESULTS FOR NUMBER OF

CUSTOMERS IN

M

/

M

/1 QUEUE

where

ρ = = Traffic intensity (ρ < 1)

(Allen, 1978; Cohen, 1985; Cox and Smith, 1961; Hillier and Lieberman, 1980; Morse, 1958; Wolff, 1989). Note: the distribution of the number of customers in system, given by Equations 7.33–7.36, is a geometric distribution with parameter 1 – ρ (see Chapter 3, Section 3.3).

Number of Customers in Queue

Number of Customers in System (including one being served)

Mean E[n] (7.29) (7.33)

Variance Var[n] (7.30) (7.34)

Probability distribution Pr{number = n}

1 – ρ2 _{(n = 0)} (1 – ρ)ρn+1 _(n≥ ₁₎

(7.31) (1 – ρ)ρn _(7.35)

Cumulative distribution Pr{number ≤n}

1 – ρn+2 _{(7.32) 1 – ρ}n+1 _(7.36)

ρ ρ 2

1−

ρ ρ 1−

ρ ρ ρ

ρ

2 2

2 1

1 + −

−

(

)

(

)

ρ ρ 1− 2

(

)

λ µ=

Arrival rate Service rate

2127_C07_frame Page 88 Friday, December 20, 2002 10:52 AM

7.13

DISTRIBUTION RESULTS FOR TIME IN

M

/

M

/1

QUEUE

where

ρ = = Traffic intensity (ρ < 1)

and

u0(t) = is the unit impulse function centered at t = 0, given by

u0(t) = and

(Allen, 1978; Cohen, 1985; Cox and Smith, 1961; Hillier and Lieberman, 1980; Kleinrock, 1975; Papadopoulos, Heavy, and Browne, 1993). Note: the distribution of the time in system, given by Equations 7.41–7.44, is an exponential distribution with parameter µ(1 – ρ) (see Chapter 4, Section 4.2).

Time in Queue

Time in System (time in queue plus service time)

Mean E[t] (7.37) (7.41)

Variance Var[t] (7.38) (7.42)

Probability density function f(t) (t≥ 0)

(1 – ρ)u0(t) + ρµ(1 – ρ)e–µ(1–ρ)t

(7.39) µ(1 – ρ)e–µ(1–ρ)t _(7.43)

Cumulative distribution function F(t) Pr{time in queue ≤ t}

1 – ρe–µ(1–ρ)t _{(7.40) 1 – e}–µ(1–ρ)t _(7.44)

ρ µ

(

)

1

1 µ

(

)

2

1

2 2

− −

(

)

(

)

µ ρ

1

1

2 2

µ

(

)

λ µ=

Arrival rate Service rate

∞

(

)

≠

(

)

  

t t

0

0 0 u t dt0 1

−∞ ∞

∫

( )

7.14

OTHER FORMULAS IN QUEUEING THEORY