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M/G/1 queue with event-dependent arrival rates

Abstract

Motivated by experiments on customers’ behavior in service systems, we consider a queueing model with event-dependent arrival rates. Customers’ arrival rates depend on the last event, which may either be a service departure or an arrival. We derive explicitly the performance measures and analyze the impact of the event-dependency. In particular, we show that this queueing model, in which a service completion generates a higher arrival rate than an arrival, performs better than a system in which customers are insensitive to the last event. Moreover, contrary to the M/G/1 queue, we show that the coefficient of variation of the service does not necessarily deteriorate the system performance. Next, we show that this queueing model may be the result of customers’ strategic behavior when only the last event is known. Finally, we investigate the historical admission control problem. We show that, under certain conditions, a deterministic policy with two thresholds may be optimal. This new policy is easy to implement and provides an improvement compared to the classical one-threshold policy.

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Notes

  1. The performance measures in the exponential case could also be obtained using a Markov chain analysis.

  2. Another way to compute the performance measures in the Erlang case is to use a matrix-geometric approach.

  3. We refer the reader to the book [12] for an overview on equilibrium behavior in queueing systems.

  4. In this case, \(E(W^+)=\overline{x}\frac{1+cv^2}{2}\left( \frac{1}{G^*(\lambda )}+\frac{\lambda \overline{x}}{1-\lambda \overline{x}}\right) +\frac{1-\lambda \overline{x}}{\lambda G^*(\lambda )}-\frac{1}{\lambda }\), and \(E(W^-)=\overline{x}\left( \frac{1+cv^2}{2}\frac{(\lambda \overline{x})^2(1-G^*(\lambda ))}{(1-\lambda \overline{x})(1-\lambda \overline{x}G^*(\lambda ))}+\frac{G^*(\lambda )+\lambda \overline{x}-1}{1-\lambda \overline{x}G^*(\lambda )} \right) \).

  5. The limit of \(E(W^+)\) corresponds to the expected remaining service time in an M/G/1 queue.

  6. [3] shows an example where deterministic policies are not optimal for the admission control in an M/G/1 queue.

  7. This particular Coxian distribution is a hypoexponential distribution.

  8. Yet, we do not claim that the two-threshold policy is optimal in this case.

Abbreviations

\(\lambda ^+\), \(\lambda ^-\) :

Arrival rates after an arrival and after a service

S :

Random variable which represents the service time duration

\(\overline{x}\) :

Expected service time

cv :

Coefficient of variation of the service time distribution; it is the ratio of the standard deviation divided by the expected value

g(.):

Probability density function of the service time

\(G^*(.)\) :

Laplace–Stieltjes transform (LST) of the service time; \(G^*(s)=\int _{0}^{\infty } g(t) e^{-s t} \, {\mathrm {d}}t\)

\(k^+\), \(k^-\) :

Thresholds on the system size or on the remaining number of service phases to accept or reject customers at arrival after an arrival or a service

\(\widetilde{p}^+\), \(\widetilde{p}^-\) :

Probabilities that an arriving customer accepts joining after an arrival or a service

\(p_t(n,r,+)\), \(p_t(n,r,-)\) :

Probability density of having n customers in the system, \(n \ge 1\), and a remaining service time of r, \(r \ge 0\), at time t after an arrival or a service

\(p(n,r,+)\), \(p(n,r,+)\) :

\(p(n,r,+)=\lim \limits _{t \rightarrow \infty } p_t(n,r,+)\), \(p(n,r,-)=\lim \limits _{t \rightarrow \infty } p_t(n,r,-)\) for \(n\ge 1\)

\(\pi _{n,+}\), \(\pi _{n,-}\) :

Stationary probability of having n customers in the system at arbitrary instants after an arrival or a service (\(\pi _{n,+}=\int _{0}^{\infty } p(n,r,+) \, {\mathrm {d}}r\), \(\pi _{n,-}=\int _{0}^{\infty } p(n,r,-) \, {\mathrm {d}}r\) for \(n\ge 1\))

\(\pi _{n}\) :

Stationary probability of having n customers in the system at arbitrary instants, \(\pi _{n}=\pi _{n,+}+\pi _{n,-}\) for \(n\ge 0\)

\(p_{n,+}\), \(p_{n,-}\) :

Stationary probability of having n customers in the system at departure instants after an arrival or a service for \(n\ge 0\)

\(p_{n}\) :

Stationary probability of having n customers in the system at departure instants, \(p_{n}=p_{n,+}+p_{n,-}\) for \(n\ge 0\)

\(\overline{r_n^+}\), \(\overline{r_n^-}\) :

Expected remaining service time seen by a customer who arrives after an arrival or a service with n customers present in the system, \(n\ge 1\).

\(\alpha _{n}\), \(\beta _n\) :

Probability that n customers arrive during a service if the service is initiated by a service completion or by an arrival, \(n\ge 0\)

\(E(Q_d)\), E(Q):

Expected number of customers in the system at departure and arbitrary instants

\(\overline{E(Q)}\) :

Service level objective on E(Q)

E(W):

Expected waiting time at arbitrary instants

\(T_s\) :

Expected throughput of served customers

\(E(W^+)\), \(E(W^-)\) :

Expected waiting time of a customer who arrives after an arrival or a service

E(R):

Expected remaining service time seen by an arriving customer at a non-empty system

\(E(R^+)\), \(E(R^-)\) :

Expected remaining service time seen by a customer who arrives after an arrival or a service

\(V_n^+(x)\), \(V_n^-(x)\) :

Value function depending on the state of the system

\(\mu _j\) :

Exponential rate of remaining service phase j in the Coxian distribution (\(1\le j \le N\))

\(r_j\) :

Probability of entering remaining service phase \(j-1\) after leaving remaining service phase j (\(1\le j \le N\)) in the Coxian distribution

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Legros, B. M/G/1 queue with event-dependent arrival rates. Queueing Syst 89, 269–301 (2018). https://doi.org/10.1007/s11134-017-9557-7

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Keywords

  • Queueing systems
  • Performance evaluation
  • M/G/1
  • Threshold policy
  • Strategic behavior

Mathematics Subject Classification

  • 90B22
  • 60K25
  • 68M20