Some considerations about the M|G|∞ queue approximation by a Markov renewal process

This is an Author's Accepted Manuscript of an article published as: FERREIRA, M.A.M. (2013)Some considerations about the M|G|∞ queue approximation by a Markov renewal process. Information Sciences and Computing. 2013(1):Art.ID ISC040713, available online at: http://www.infoscicomp.com/

Research Article

Some Considerations about the

||∞ Queue

Approximation by a Markov Renewall Process

Manuel Alberto M. Ferreira

Instituto Universitário de Lisboa (ISCTE – IUL), BRU - IUL, Lisboa, Portugal Correspondingauthor: Manuel Alberto M. Ferreira; E-mail: [email protected] Received 04 June 2013; Accepted 11 July 2013

Abstract:SomeM|G|∞ queue systems interesting quantities values approximations, obtained

through the consideration of an adequate Markov renewal process, are presented and studied.

Keywords:M|G|∞; Markov renewal process; queue.

1. Introduction

In the M|G|

∞

queue system the customers arrive according to a Poisson process at rateλ, receive a service which time is a positive random variable with distribution function G

()

. and mean α and, when they arrive, find immediately an available server. Each customer service is independent from the other customers’ services and from the arrivals process. The traffic intensity isρ =λα.

A suggestion to obtain approximate results for these systems, when the exact ones are steel not known, is to use a Markov renewal process, see (1-2).

Along this work, some of the approximations so obtained are reviewed.

2. Sojourn Time Mean Value in State k

For theprocess referred above, the sojourn time mean value in state1,  = 0,1, …is given by  =   1 −  ∞     ,  = 0,1, … 2.1 ∞ ! . 1

The state of the M|G|∞ queue in instantt is the number of customers being served in the system at instantt.

Proposition 2.1

! = "_ 2.2.∎

Obs: The sojourn time mean value in state 0, does not depend neither on . 

nor on the arrivals process. It depends only on the arrivals process.

Proposition 2.2

 ≤"_ ,  = 0,1, … 2.3.

Dem: It is enough to note that& 1 −  ≤ 1_∞ .∎

Obs: So, the sojourn time mean value in any state does not exceed the one of the state

0. Put

'! = 1_{( 2.4.}

Proposition 2.3

 ≤ *_{2/2 + 1 , ,  = 1,2, … 2.5}+,-+ 1 being+_, the service coefficient of variation.

Dem: Using the Schwartz’s inequality, -≤ ∞ -

!  1 "2343 ∞ 5 6 7 -  = ∞ ! " -689 : 1 −  ∞  ; -  = ∞ ! -6"89-6 8 - +,-+ 1 689<=_>89<= --?" ≤  @A8?" --?" since, see (3),  B 1 − ∞  C D  = ∞ ! E -2 +,-+ 1 D"_FD" EE + 1 withFD ≤ 2, E = 0,1, … 2.6. ∎ Obs: Define '" = *_{2/2 + 1 2.7.}+,-+ 1 Proposition 2.4 If  ≥" N/+,-+ 1 −"-, '" ≤ '!. Dem: *_{2/2 + 1 ≤}+,-+ 1 1_{( ⇔2/2 + 1 ≤}+,-+ 1 _/1_- ⟺ /+,-+ 1 ≤ 4 + 2 ⟺  ≥1_{4 /+,}-_{+ 1 −}1 2 . ∎

Proposition 2.5  ≤ +, -_{+ 1}  + 1 ,  = 1,2, … 2.8. Dem:  ≤   1 −  ∞      ≤_1₂ -∞ ! +, -_{+ 1} 2"  + 1 = +, -_{+ 1}  + 1 after (2.6).∎ Obs: Define '- = +, -_{+ 1}  + 1 2.9. Proposition 2.6 If  ≥ /+_,-+ 1 − 1, '_- ≤ '_!. Dem: +_{ + 1 ≤},-+ 1 1_{( ⟺ /+},-+ 1 ≤  + 1 ⟺  ≥ /+,-+ 1 − 1. ∎ Proposition 2.7 If  ≤ 2/+_,-+ 1 − 1, '_" ≤ '_-. Dem: '" '- = S @A8?" -T-?" @A8?" ?" = U+,-+ 1 + 1 +,-+ 1U2/2 + 1 =  + 1 U+,-+ 1U2/√2 + 1 ≤ *_2/+ + 1 ,-+ 1 ;  + 1 2/+,-+ 1 ≤ 1 ⇔  + 1 ≤ 2/+, -_{+ 1 ⟺ } ≤ 2/+,-+ 1 − 1. ∎ Proposition 2.8 If  ≥ 4/+_,-+ 1 − 1, '_- ≤ '_". Dem: '" '_- = _U2/+,- + 1_{+ 1√2 + 1}≥ _U4/+,- + 1_{+ 1√ + 1}= *4/+ + 1_,-+ 1 ;4/+ + 1_,-+ 1 ≥ 1 ⟺  ≥ 4/+,-+ 1 − 1. ∎

The Propositions 2.4, 2.6, 2.7 and 2.8 lead to the following upper bounds choice for

A)YZ_[\+ & >\ ^  <1_{4 /+},-+ 1 −1₂  ≤ 1₍ 1 4 /+,-+ 1 −12 ≤  ≤ 2/+,-+ 1 − 1  ≤ *2/2 + 1+,-+ 1 2/+,-+ 1 − 1 <  < 4/+,-+ 1 − 1  ≤ `E a*<sub>2/2 + 1 , </sub>+,-+ 1 +, -<sub>+ 1</sub>  + 1 b  ≥ 4/+,-+ 1 − 1 ≤ +, -<sub>+ 1</sub>  + 1 B)& \< /Z[\+ & ≤ \ ^  = 1 " ≤ `E a*+,-_{6/ , }+ 1 +, -_{+ 1} 2 b  = 2,3, …  ≤ +, -_{+ 1}  + 1

C)YZ_[\+ & ≤&

\

 ≤ +, -_{+ 1}

 + 1 ,  = 1,2, …

Proposition 2.9: If the service time distribution is NBUE

 ≤ _{ + / ,  = 1,2, … 2.10}

Dem: It is enough to note that if the service time is NBUE with mean ,  1 −_>∞

 ≤  ∞ dc

> , for any F ≥ 0. ∎

Obs: If the service time is NWUE with mean ,  1 − _>∞ ≥  ∞ dc

> , for

any F ≥ 0 and

 ≥ _{ + / ,  = 1,2, … 2.11.}

 ≥ e"

8

f_g88dhfi j

-j-( + 2 ,  = 1,2, … 2.12 Being j_- and j_k the 2nd and the 3rd.  moments around the origin

Dem: If the service time2 is IMRL

1 − ∗_{ = 1 −}" 6 1 − mm = 3 !  "2n4n ∞ c 6 ≥ 8d_g838_fg88dhf?" . So, ≥  e  8d g88f_g88dhf?"_i ∞ !  = e"8f_g88dhfi ∞ o?8d_g8p_ ! = e"8_fd g88hfi −1 ( + -6_h 8 1 o?g88dp7 ! ∞ = e"8f_g88dhfi_. j -j-( + 2 . ∎

Proposition 2.11: If the service time distribution is DFR3

 ≥ q

=<rA8

8 s 

 + / ,  = 1,2, … 2.13.

Dem: If the service is DFR1 −  ≥ cdrA88?=8 .

So,  ≥ _1 B  c drA88?=8 ∞  C   = q=<rA8₈ s  ∞ !  _B ∞ c_d  C  = ∞ ! q=<rA88 s   + / . ∎

Note: It is not known an expression to the sojourn time value in state kfor the M|G|

∞

queue systems, with the exception of a) = 0, for every . , being

_! =1_{( 2.14}

2_∗ ="

6 1 − mm 3

! is the service time equilibrium distribution.

3_{For more details about NBUE (New Better than Used in Expectation), NWUE (New Worse than Used}

in Expectation), IMRL (Increasing Mean Residual Life) and DFR (Decreasing Failure Rate) distributions, important in reliability theory, see (4).

b) Everyk, for exponential service time, where

 = _?T6 ,  = 0,1, … 2.15.

In the same circumstances, the Markov renewal process supplies the same results: in fact (2.14) is equal to (2.2) and if  = 1 − dc,  ≥ 0 in (2.1) it is obtained (2.15).

-The bounds given by (2.10), (2.11), match the exact value given by (2.15). The expression (2.13) is coincident with (2.15) for +_, = 1.

3. State 0 Recurrence Mean Time

For the Markov renewal process, the state 0 mean recurrence time4 is given by

j! ="_t1 + ∑∞xy"∏xy"_"ww9₉z 3.1. Proposition 3.1 If / ≤_@" A8?", j! ≤ {|}rA8~=  3.2

Dem: To use an upper bound of  in (3.1) it is necessary to certify that it is lesser

than "

 . The condition /+,-+ 1 ≤ 1, due to Proposition 2.6, guaranties that

'-fulfills that request for  ≥ 1.

So j_! ≤ " 1 + ∑ ∏ |}rA8~= 9~= "|}rA8~=_9~= x y" ∞

xy"  ="_11 + ∑xy"∞ ∏x_y"?"T}@T}@A8?"_A8_?"7 ≤

" B1 + ∑ :T}@A8?";€ x! ∞ xy" C ={|}rA8~=_ . ∎

Obs: For the M|G|

∞

queue systems

j! ={

|

(3.3).

So, in these conditions, the relative error arising from considering (3.2) instead of(3.1) is {|}rA8~=  − {|  {|  = T@A8− 1 ≤ rA8~=rA8 _{− 1 < − 1.} But rA8 rA8~=− 1 ≤ ‚ ⟺ @A8 @A8?"≤ log‚ + 1 ⟺ +, - _≤ †‡ˆ‰?" "†‡ˆ‰?".

That is: if /+_,-+ 1 ≤ 1, the relative error arising from taking the bound given by (3.2) instead of the true value given by (3.3) for j_! is such that:

4

a)Š ≤ rA8 rA8~=_{− 1,} b)Š = 0 if+_,- = 0, c)Š < − 1, d)Š ≤ ‚‚ < − 1since+_,- ≤ †‡ˆ‰?" "†‡ˆ‰?".

So, requesting that Š is lesser than a given r, it results a criterion to measure the goodness of the  approximation by '_- for a certain+_,-, since /+_,-+ 1 ≤ 1.

Being B the M|G|

∞

queue busy period length, see (5-6),

' = T_{( 3.4.}− 1

For the Markov renewal process, since/+_,-+ 1 ≤ 1,

' ={|}rA8~=_ " 3.5.

Now, the relative error own to take (3.5) instead (3.4), is

‘|}rA8~=<= ’ ‘|<=’ ‘|<= ’ ={|}rA8~=_{|_"{| ={ |rA8_" "{<| ≤{ rA8 rA8~=_" "{<| <_"{{"<|. So a)“ ≤{ rA8 rA8~=_" "{<| , b)Š = 0 if+_,- = 0, c)“ < {" "{<|, d)“ ≤ ‚ o‚ < {" "{<|p since +,- ≤ †‡ˆ‰"{ <|_?" "†‡ˆ‰"{<|_?".

So, the bounds for “ are greater than the obtained to Š. Then it is preferable to use a

criterion based on Š than on “, to measure the goodness of the approximation of 

by '_-.

4. Mean Number of Entries in State k between Two Entries in State 0

For the Markov renewal process, the mean number of entries in state kbetween two entries in state 0 is

Proposition 4.1 If / ≤ " @A8?" ”≤  + 1T 9<=_}@A8?"9<= ! ,  = 1,2, … 4.2.∎

Obs.: Values for ”,  = 1,2, … for theM|G|

∞

queue system are not known,

From (2.2), (2.9) and (4.2) it follows that

”≤ 6T

9<=_}@A8?"9

! ,  = 0,1, … (4.3) Since/+_,-+ 1 ≤ 1.

For the M|G|

∞

queue system

”= 6T 9<= ! ,  = 0,1, … (4.4). But d|9<=}rA8~=9 9! d|9<=9! d|9<= 9!

=+_,-+ 1− 1, that is null for +_, = 0 or = 0 and

increases with k if +_,- > 0.

Note: For / ≤ 1 –E+_,- = 0 the Markov renewal process suppliesthe following

results: a)_! = " , b) ≤_?"6 ,  = 1,2, … c)j_! ≤{| , d)' ≤{|"  , e)” ≤  + 1T9<= ! ,  = 1,2, … f) ” =6T9<= ! ,  = 0,1, …

So the value obtained for _! coincides with the M|G|

∞

one. And the bounds obtained forj_!,' and ” coincide with the true values for the same M|G|

∞

quantities.But the bounds obtained for  and ”coincide with the true value obtained when the

service time distribution is exponential and the traffic intensity is 1. In opposition, the bound got for  cannot coincide with the one given by 2.15for / < 1. So it is

excluded the hypothesis of having an expression for  independent from the service

time distribution and equal to the one given by 2.15. Then, only rarely the Markov

renewal process gives values for j_!, ' and ” identical to the M|G|

∞

ones. If /+_,-+ 1 ≤ 1 it is possible, after the Markov renewal process, to get upper bounds for the M|G|

∞

system quantities j_!, ' and ”. So it is admissible to consider that at least '_-, beyond being a  upper bound for the Markov renewal process, plays also the same role for the M|G|

∞

queue.

Note still that if +_,- = 0, for the Markov renewal process

 =  t1 −₆z  ,  = 0,1 … 4.5 6 ! . So, for  ≥ 1,  ≤  o1 −  6p   6 ! =1?"6 o1 −  6p ?" 7 ! 6 =_?"6 , That is  ≤ 6 ?",  = 1,2, … . But, requesting that 6

?"≤" ⟺  ≥ / − 1 that leads to / − 1 ≤ 0 ⟺ / ≤ 1. -_" =  o1 − 6p  = t− {<’5  o1 −  6pz_! 6 − 6 !  −_!6 {<’5_ o−₆"p ="_−"_T _{ =}" −"Tt{ <’5 z_! 6 = "_−"_To−{<|_ +"_p 6 ! . So, " = T?{ <|_" T8 (4.6).

And, integrating by parts,

?" ="_−?"_T , k=1,2,…(4.7).

With (4.6) and (4.7) it is possible to obtain ,  = 1,2, … for +_,- = 0 and it is possible to conclude that, in this case, (2.15) does not hold.

Proposition 4.2: If the service time distribution is NBUE

a)j_! ≤{_|,

b)' ≤{|"  , c)” ≤  + 1T9<=

d) ” ≤6T_!9<=,  = 0,1, … ∎

Obs: The bounds obtained for j_!, ' and ” coincide with the true value of

these quantities for the M|G|

∞

queue. If the service time distribution is NWUE a)j_! ≥{| , b)' ≥{|"  , c)” ≥  + 1T9<= ! ,  = 1,2, … d) ” ≥6T9<= ! ,  = 0,1, …

with a comment identical to the one in the case NBUE. So it is admissible that 6

?T,  = 0,1, …is an upper bound(lower bound) for the true value of in the M|G|

∞

queue systems in the case ofNBUE (NWUE) service time distributions.

After (2.1) and integrating by parts

?" =  1 "2343 ∞ 5 6 7 ?"  = B−{<’5_ q "23435∞ 6 s ?" C ! ∞ ∞ ! −  −{ <’5   + ∞ ! 1 1 "23435∞ 6 7 _2" 6  = " − ?" T  1  "2343∞ 5 6 7  }1 −  ≥"_− ∞ ! ?" T . So ?" ≥1_{( −} + 1_/  ,  = 1,2, … 4.8.

Note:According to (4.7), when the service time is constant, the equality holds in (4.8).

5. Sojourn Time in State k Distribution

The sojourn time in sate k distribution function for the Markovrenewal process is

— = 1 −  1 −  ∞     ,  ≥ 0,  = 0,1, … 5.1. Evidently,

Proposition 5.1

— ≥ 1 − ,  ≥ 0,  = 0,1, … 5.2. ∎

Proposition 5.2: If the service time distribution is exponential

— = 1 − 

=

d?T,  ≥ 0,  = 0,1, … 5.3. ∎

Obs.: This result is coincident with the one known for theM|G|

∞

queue.

Proposition 5.3

—! = 1 − ,  ≥ 0, 5.4. ∎

Obs.: Result obvious for any M|G|

∞

queue and for any queue with Poisson arrivals

process.

Proposition 5.4: If the service time distribution is NBUE

— ≥ 1 − 

=

d?T,  ≥ 0,  = 0,1, … 5.5. ∎

Obs.: As emphasized before, (5.5), beyond supplying a lower bound for — in the

Markov renewal process, also gives a lower bound for that quantity in the M|G|

∞

system for the case of NBUE service time.

If the service time distribution is NWUE

— ≤ 1 − 

=

d?T,  ≥ 0,  = 0,1, … 5.6

And it is pertinent a comment analogous to the former one with the change of lower bound by upper bound.

Proposition 5.5: If the service time distribution is IMRL

— ≤ 1 − ?e

8d

g88f_g88 d hf?"i_{,  ≥ 0,  = 0,1, … 5.7∎} Proposition 5.6: If the service time distribution is DFR

— ≤ 1 − 

=

d?T?=<rA88 ,  ≥ 0,  = 0,1, … 5.8. ∎

Conclusions

When analytical exact results are not available, numerical methods are used to try to find approximations for the interesting quantities under study. It is what is done in this

work for the M|G|

∞

queue, trying to approximate it for a Markov renewal process.An

Still another is to determine service time distributions for which it is possible to determine the most of the interesting quantities for the M|G|

∞

queue. This is made solving differential equations induced for the study of the transient behaviour, see (8-10).

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Copyright © 2013 Manuel Alberto M. Ferreira. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.