Analysis of queueing systems with customer interjections

Abstract

In this paper we study queueing systems with customer interjections. Customers are distinguished into normal customers and interjecting customers. All customers join a single queue waiting for service. A normal customer joins the queue at the end and an interjecting customer tries to cut in the queue. The waiting times of normal customers and interjecting customers are studied. Two parameters are introduced to describe the interjection behavior: the percentage of customers interjecting and the tolerance level of interjection by individual customers. The relationship between the two parameters and the mean and variance of waiting times is characterized analytically and numerically.

Appendix: An efficient algorithm for the M/M/1 case

The computation approach taken here is similar to the one used in Sect. 3. Since this is a special case, some limits are obtained explicitly and the approximations to moments of waiting times are significantly simpler. Based on the first passage time analysis [16], we develop a stable algorithm for computing the mean and variance of W _n (η _I ,η _C ). Denote by T _n the first passage time for a customer in position n to move to position n−1 in queue, which is the first passage time from level n to n−1 for the Markov chain Q _a defined in Eq. (2.2).

Similar to the proof of Lemma 2.2, it can be shown that T _n is stochastically larger in n (also see [3]). Thus, E[T _n ] and $E[T_{n}^{2}]$ are increasing in n. Let T _∞=lim _n→∞ T _n . The variable T _∞ can be considered as the length of the busy period of an M/M/1 queue with arrival rate λη _I and service rate μ. It is readily seen that

$$ \everymath{\displaystyle} \begin{array}{rcl} E \bigl[e^{ - sT_{n}}\bigr] &=& \frac{\mu + \lambda \eta_{I}(1 - (1 - \eta_{C})^{n})E[e^{ - sT_{n + 1}}]E[e^{ - sT_{n}}]}{s + \mu + \lambda \eta_{I}(1 - (1 - \eta_{C})^{n})}, \quad n \ge 1; \\[12pt] E \bigl[e^{ - sT_{\infty}} \bigr] &=& \frac{\mu + \lambda \eta_{I} ( E[e^{ - sT_{\infty}} ] )^{2}}{s + \mu + \lambda \eta_{I}}. \\ \end{array} $$

(A.1)

E[T _∞] and $E[T_{\infty}^{2}]$ are finite and are given by:

$$ \everymath{\displaystyle} \begin{array}{rcl} E[T_{\infty} ] &=& \frac{1}{\mu - \lambda \eta_{I}}, \\[14pt] E \bigl[T_{\infty}^{2}\bigr] &=& \frac{2E[T_{\infty} ](1 + \lambda \eta_{I}E[T_{\infty} ])}{\mu - \lambda \eta_{I}} = \frac{2\mu}{ ( \mu - \lambda \eta_{I})^{3}}. \end{array} $$

(A.2)

The following relationships of {T _n ,n≥1} can be shown routinely:

(A.3)

The second term on the third line of Eq. (A.3) is zero since ρη _I <1 and E[T _∞] is finite. Similarly, we can obtain

$$ \everymath{\displaystyle} \begin{array} {rcl} E \bigl[T_{n}^{2}\bigr] &=& \frac{2E[T_{n}] + \lambda \eta_{I}(1 - (1 - \eta_{C})^{n}) ( E[T_{n + 1}^{2}] + 2E[T_{n}]E[T_{n + 1}] )}{\mu} \\[6pt] &=& \frac{2}{\mu} \sum_{k = 0}^{\infty} (\rho \eta_{I})^{k}E[T_{n + k}] \bigl( 1 + \lambda \eta_{I}\bigl(1 - (1 - \eta_{C})^{n + k} \bigr)\\[6pt] &&{}\times E[T_{n + k + 1}] \bigr) \Biggl( \prod_{j = 0}^{k - 1} \bigl(1 - (1 - \eta_{C})^{n + j}\bigr) \Biggr) ; \\[6pt] \mathit{Var}[T_{n}] &=& \biggl( \frac{1}{\mu} + \frac{\lambda \eta_{I} (1 - (1 - \eta_{C})^{n})E[T_{n + 1}]}{\mu} \biggr)^{2} + \frac{\lambda \eta_{I}(1 - (1 - \eta_{C})^{n})}{\mu} E\bigl[T_{n + 1}^{2} \bigr]. \end{array} $$

(A.4)

By Eqs. (A.3) and (A.4), it is easy to obtain the following result.

Lemma A.1

The functions E[T _n ], $E[T_{n}^{2}]$, and Var[T _n ] are non-decreasing functions of η _I and η _C . In addition, the three functions are convex in η _I .

Since W _n (η _I ,η _C )=T ₁+T ₂+⋯+T _n , we have

$$ \everymath{\displaystyle} \begin{array}{rcl} E\bigl[W_{n}(\eta_{I}, \eta_{C})\bigr] &=& E[T_{1}] + E[T_{2}] + \cdots + E[T_{n}] = E\bigl[W_{n - 1}(\eta_{I}, \eta_{C})\bigr] + E[T_{n}]; \\[6pt] E\bigl[W_{n}^{2}(\eta_{I},\eta_{C}) \bigr] &=& \sum_{j = 1}^{n} E\bigl[T_{j}^{2} \bigr] + 2\sum_{1 \le i < j \le n} E[T_{i}]E[T_{j}] \\[12pt] &=& E\bigl[W_{n - 1}^{2}(\eta_{I},\eta_{C}) \bigr] + E\bigl[T_{n}^{2}\bigr] + 2E\bigl[W_{n - 1}(\eta_{I},\eta_{C})\bigr]E[T_{n}]. \end{array} $$

(A.5)

Since the variance of W _n (η _I ,η _C ) is given by $\mathit{Var}[W_{n}(\eta_{I},\eta_{C})] = \sum_{i = 1}^{n} \mathit{Var}[T_{n}]$, we obtain the following lemma.

Lemma A.2

The functions E[W _n (η _I ,η _C )] and Var[W _n (η _I ,η _C )] are non-decreasing in both η _I and η _C . In addition, the two functions are convex in η _I .

To approximate the mean and variance of W _n , choose sufficiently large N and set E[T _N ]=E[T _∞] and $E[T_{N}^{2}] = E[T_{\infty}^{2}]$. Then E[T _n ] and $E[T_{n}^{2}]$ can be computed by using the formulas in Eqs. (A.3) and (A.4) for n≤N. Using the formulas in Eq. (A.5), E[W _n ] and $E[W_{n}^{2}]$ can be computed for n≥1.

Note that, by the monotonicity property of {T _n ,n≥0}, we have E[W _n ]≤nE[T _∞] and $E[W_{n}^{2}] \le nE[T_{\infty}^{2}] + n(n - 1)(E[T_{\infty} ])^{2}$. Thus, we can choose large enough N so that the error in computing the first two moments of waiting times, such as E[W _[N](η _I ,η _C )] and E[(W _[N](η _I ,η _C ))²], can be smaller than any given positive number.