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IQueueing Systems with Fractional Number of Servers

KeskusteluaiheitaDiscussion Papers

21.3.2012

No 1268

* ETLA – The Research Institute of the Finnish Economy, [email protected]

** ETLA – The Research Institute of the Finnish Economy, [email protected]

Valeriy Naumov * – Olli Martikainen**

Queueing Systems withFractional Number of

Servers

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ETLA Keskusteluaiheita – Discussion Papers No 1268II

ISSN 0781–6847

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1Queueing Systems with Fractional Number of Servers

Contents

Abstract 2

1 Introduction 3

2 M/M/s queue 3

3 Extended M/M/s queue 5

4 Conclusion 7

References 9

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ETLA Keskusteluaiheita – Discussion Papers No 12682

Abstract

In this paper we introduce multi-server queueing systems that can be considered as extensions of conven-

tional M/M/s queue to fractional number of servers. We show that the extended Erlang’s delay function

can be used to calculate delay probabilities for such systems. This approach enables the delay analysis in

networks with fractional number of servers in the nodes using classical methods.

Key words: Erlang’s delay function, multi-server system, state dependent service times

JEL: C61, C62, C68

Tiivistelmä

Tässä artik kelissa esittelemme monen palvelimen jonojärjestelmän, jossa traditionaalinen M/M/s jonokä-

site laajennetaan murtolukumäärälle palvelimia. Näytämme, että yleistettyä Erlangin viivefunktiota voi-

daan käyttää tällaisen järjestelmän viivetodennäköisyyksien laskemiseen. Esitetty menetelmä mahdollis-

taa viiveiden laskennan klassisia menetelmiä käyttäen yleistetyissä jonojärjestelmissä, joiden solmuissa

on murtolukumäärä palvelimia.

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1 Introduction

We introduce multi-server queueing systems that can be considered as extensions of

conventional M/M/ s queue to fractional number of servers. We show that the extended

Erlang’s delay function can be used to calculate delay probabilities for such systems. This

approach enables the delay analysis in networks with fractional number of servers in the

nodes using classical methods.

Erlang’s loss function and Erlang’s delay function have been widely used for the analysis of

multi-server systems since their appearance in (Erlang, 1917). These functions give the

blocking probability for systems without waiting places and delay probability for systems

without losses. Analytical continuation extends these functions to non-integral value of the

number of servers (Bretschneider, 1973), (Jagers and van Doorn, 1991). Extended Erlang’s

loss function is used for the analysis of complex systems with losses by teletraffic

engineering methods like equivalent random traffic and Hayward’s approximation (Iversen,

2011). Equivalent random traffic for queues method as well as optimal server allocation

requires calculation with extended Erlang’s delay function (Nightingale, 1976), (Down and

Karakostas, 2008), (Naumov and Martikainen, 2011). But, to the author's knowledge, there

are no descriptions of systems which can be interpreted as multi-server system with fractional

number of servers. In this paper we introduce a queueing system, in which the delay

probability is given by extended Erlang’s delay function, and which for integer value of

parameter s operates as conventional multi-server queueing system.

2 M/M/ s queue

Consider queueing system with s servers and infinite number of waiting places. Assume that

arrival process is Poisson with arrival rate λ , service times are exponentially distributed withservice rate μ and s

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⎟ ⎠

⎞⎜⎝

⎛ −

+

⎟ ⎠

⎞⎜⎝

⎛

−=

∑−

= ρ

ρ ρ

ρ

ρ

ρ

s

s

sk

s

s

sC

s s

k

k

s

s

!!

!)(

1

0

. (2)

Erlang’s delay function )( ρ sC can be expressed through Erlang’s loss function )( ρ s B as

))(1(

)()(

ρ ρ

ρ ρ

s

s s

B s

sBC

−−= , (3)

where

1

0!!

)(

−

= ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ = ∑

s

k

k s

sk s

B ρ ρ

ρ .

(4)

For traffic engineering purposes, Bretschneider has proposed in (Bretschneider, 1973) an

analytic continuation of Erlang’s loss function to non-integral values of s by

dt et

e B

t s

s

s−∞

−

∫=

ρ

ρ ρ

ρ )( ,(5)

which can be rewritten as (Jagerman, 1974):

1

0)1()(

−∞ −⎟ ⎠

⎞⎜⎝

⎛ += ∫ dt t e B st

s ρ

ρ ρ . (6)

For all integer s this function coincides with (4). Similar analytic continuations of Erlang’s

delay function, which for all integer s coincides with (2), have been proposed in (Jagers and

van Doorn, 1991) by

1

0

1)1()(−∞ −−⎟ ⎠

⎞⎜⎝

⎛ += ∫ tdt t eC st

s ρ

ρ ρ . (7)

It is easy to prove that formula (3) also remains valid for functions (6) and (7) (see, e.g.,

Karsten et al. 2011).

For calculation of extended Erlang’s delay function )( ρ sC

we use formula (3) in which

extended Erlang’s loss function )( ρ s B is computed in Matlab environment (Martinez and

Martinez, 2002). We rewrite (5) as

)(1

)()(

1

1

ρ

ρ ρ

+

+

−=

s

s s

F

f B , (8)

where )( x f γ

and )( x F γ are the probability density function and the cumulative distribution

function of the standard gamma distribution respectively,

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xe xa

x f −−

Γ= 1

)(

1)( γ γ , dt et

a x F t

x −−∫Γ

=0

1

)(

1)( γ γ .

In Matlab environment these functions can be computed as )1,,(gampdf )( γ γ x x f = and

)1,,(gamcdf )( γ γ x x F = .

3 Extended M/M/ s queue

In this section we describe a family of multi-server systems with Poisson arrival processes

and state dependent service rates, which can be interpreted as extensions of the queueing

system M/M/s to non-integral value of s. Let )( x R be continuous function, which is positive

for 0> x and satisfies:

⎣ ⎦ ⎣ ⎦ 1)( + x , (9)

where ⎣ ⎦ x is the integer part of x. Let also s be a positive number, ⎣ ⎦ sn = , and )( s Rr = .

Consider a queueing system with Poisson arrival process, 1+n servers, and infinite number

of waiting places (see Figure 1). Servers 1,2,…, n have same service rate μ , while service

rate of the server n+1 depends of the customer presence in the queue. If the queue is empty,

server n+1 has service rate μ )( nr − , otherwise it has service rate μ )( n s − . We call

servers 1,2,…, n as fast servers and the server n+1 as slow server, since its service rate is

always less that μ . Note, that for 10

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may be busy only if there are no free fast servers. As soon as a server becomes free and the

queue is not empty a customer is selected from the queue and dispatched to the free server.

If s is close to n+1 then service rate at the slow server is close to μ . It processes customers as

fast server and the system behaves as M/M/n+1 queue. If 1> s is close to n then the slow

server unlikely has enough time to finish service: when a fast server finishes its service the

work processed at the slow server is interrupted and continued at the fast server. In this case

the slow server is used as additional waiting place and the system behaves as M/M/n queue.

Let λ be arrival rate, and s

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1) ))()()(()(

))()()(()()(

1 ρ ρ ρ ρ ρ ρ

ρ ρ ρ ρ ρ ρ

s B s sn B s s B

s B snn B sn B s s R

+−−+

+−−+=

)(

)(

)(

)(

ρ

ρ ρ

ρ

ρ ρ

n B

sC s

s B

sC n

s

+−

+−

= ; (15)

2) s s R =)(2 ; (16)

3) ρ δ δ

δ

)1()(3

−++=

c

cn s R , n s −=δ . (17)

Case 1. If we equalise right sides of (1) and (13), for the function )( s R we get formulas (15).

In this case for any positive s the mean response time in the described system is given by

formula (1), and for the delay probability we have )()(),(1 ρ ρ s s s R C C = . We may say that the

system with )()( 1 s R s R = is a truly extension of the system M/M/s to non-integral value of s.

Case 2. In the simplest case with s s R =)( the service rate of the slow server does not

depend of the number of customers in the queue. The number of customers in the system is

the birth-and-death process with birth rates λ λ =k and death rates given by

⎩

⎨⎧

+≥

≤≤=

,1,

,1,

nk s

nk k k

μ

μ μ

and for the mean residence time in the system we have

⎟ ⎠

⎞⎜⎝

⎛ −

−= δ

ρ ρ

s

sn pC s s )()(, .

Case 3. This case does not require calculation of gamma distribution but )(),(3 ρ s s RC

gives

better

approximation to the delay probability )( ρ sC then )(, ρ s sC . Figures 2 and 3 show

delay probability )(),( ρ s s R j

C

for functions )(1

s R , )(2

s R , )(3

s R , and ρ/ s=0.1, 0.5, 0.9.

4 Conclusion

In this paper we introduced a family of multi-server queueing systems that can be considered

as extensions of conventional M/M/ s queue to fractional number of servers s. One member of

the family, specified by the function (15), has the mean residence time given by formula (1)

with extended Erlang’s delay function. Other family members can be used for approximate

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calculation of delays in multi-server queues with fractional number of servers. Each system

has Poisson arrival process and state-dependent service rates. Therefore our approach

enables the analysis of open and closed networks with nodes having fractional number of

servers using well known methods (Lazowska, 1984).

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References

A.K. Erlang. Løsning af nogle Problemer fra Sandsynlighedsregningen af Betydning for de automatiske Telefoncentraler, Elektrotek nikeren, 13, 5–13,1917.

G. Bretschneider. Extension of the equivalent random method to smooth traffics, Extension of the

Equivalent Random Method to Smooth Traffics. Proc. of 7th International Teletraffic Congress, Stockholm,

411/1–9, 1973.

D.L. Jagerman. Some properties of the Erlang loss function. Bell System Technical J., 53(3), 525–551, 1974.

L. Kleinrock. Queueing Systems, Volume 1. Wiley, New York, 1975.

D.T. Nightingale. Computations with smooth traffics and Wormald chart, Proc. of 8th International

Teletraffic Congress, Sydney, 145/1–7, 1976.

M. Raiser. Mean value analysis and convolution method for queue-dependent servers in closed queueing

networks, Performance Evaluation, 1(1), 7–18, 1981.

E.D. Lazowska et.al. Quantitative system performance computer system analysis using queueing network

models, Prentice-Hall, London, 1984.

A.A. Jagers and E.A. van Doorn. Convexity of functions which are generalizations of the Erlang loss

function and the Erlang delay function. SIAM Review , 33(2), 281–282, 1991.

W.L. Martinez and A. R. Martinez. Computational Statistics Handbook with MATLAB, Chapman & Hall/CRC

Press, New York, 2002.

D.G. Down and G. Karakostas. Maximizing throughput in queueing networks with limited flexibility,

European J. of Operational Research, 187(1), 98–112, 2008.

V. Naumov and O. Martikainen. Method for throughput maximization of multiclass network with flexible

servers. ETLA Discussion paper 1261, The Research Institute of the Finnish Economy, Helsinki, 1–27, 2011.

V.B. Iversen. Teletraffic Engineering and Network Planning, DTU Course 34340, Technical University of

Denmark, Lyngby, 2011.

F. Karsten, M. Slikker, and G.-J. van Houtum. Resource pooling and cost allocation among independent

service providers, Working Paper 352, Beta Research School for Operations Management and Logistics,

Eindhoven, 2011.

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Figure 2. Probability of delay )(),( ρ s s RC

for 50 ≤< s ,

R 1 , R 2 , R 3 .

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0 1 2 3 4 5

ρ=0.1s

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

ρ=0.5s

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0 1 2 3 4 5

ρ=0.9s

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Figure 3. Probability of delay )(),( ρ

s s R

C

for 2520 ≤≤ s ,

R 1 , R 2 , R 3 .

1.0E-17

1.0E-16

1.0E-15

1.0E-14

1.0E-13

20 21 22 23 24 25

ρ=0.1s

0.001

0.002

0.003

0.004

20 21 22 23 24 25

ρ=0.5s

0.50

0.51

0.52

0.53

0.54

0.55

0.56

20 21 22 23 24 25

ρ=0.9s

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